-THE ELL EQUATION THE PELL EQUATION BY EDWARD EVERETT WHITFORD INSTRUCTOR IN MATHEMATICS IN THE COLLEGE OF THE CITY OF NEW YORK SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY, IN THE FACULTY OF PURE SCIENCE, COLUMBIA UNIVERSITY E. E. WHITFORD COLLEGE OF THE CITY OF NEW YORK NEW YORK, U. S. A. 1912 THE PELL EQUATION. By EDWARD EVERETT WHITFORD. New York: E. E. Whitford, College of the City of New York, New York, U. S. A. Pp. iv + 193. $1.00, postpaid. A history of the equation x2 - Ay2= 1, table of solutions from A = 1,501 to A = 1,700, bibliography with references to over 300 authors, table of continued fractions for i/A. PRESS OF THE NEW ERA PRINTING COMPAN1 LANCASTER. PA. CONTENTS. PAGE Origin of the name Pell Equation................. 1 D efinition..................................... 3 Relation to square root approximation............ 4 The most ancient Hindu solutions................ 6 The most ancient Greek solutions................ 9 Theon of Smyrna............................... 13 A rchim edes.................................... 15 Heron of Alexandria............................ 21 D iophantus.................................... 22 The H indus.................................... 26 The Arabs..................................... 39 Europeans before Fermat........................ 41 Fermat, Brouncker and Wallis.................. 46 E u ler......................................... 59 Relation to the general indeterminate equation of the second degree................................ 64 L agrange...................................... 71 Relation to the theory of quadratic forms......... 76 Gauss...................................... 77 Dirichlet...................................... 79 Relation to circle division....................... 88 Relation to elliptic functions..................... 91 Relation to units of a quadratic domain........... 92 Relation to hyperbolic functions.................. 93 Classes of fundamental solutions................. 93 Tables of solutions............................. 95 Table of the fundamental solutions of the equations x2- Ay2 = = 1 from A = 1,501 to A = 1,700... 102 Bibliography................................. 113 Appendix A, table of continued fractions for JIA from A = 1,501 to A = 2,012.................. 162 Index of names............................. 191 iii VITA. Edward Everett Whitford was born in Brookfield, N. Y., January 31, 1865; graduated from Brookfield Academy in 1881; received the degree of A.B. from Colgate University in 1886 and of A.M. in 1890. He taught in Colby Academy, New London, N. H., Keystone Academy, Factoryville, Pa., Shamokin (Pa.) High School, Commercial High School and Pratt Institute, Brooklyn, N. Y. He was principal of Brookfield High School, 1900-1. He is now instructor in mathematics in the College of the City of New York with which institution he has been connected since 1905. He has been a graduate student in Columbia University since February, 1904, and is a member of the American Mathematical Society. The writer takes this opportunity of expressing his thanks to Professor David Eugene Smith for fruitful suggestions and able and helpful criticism. 523 WEST 151ST STR., NEW YORK, December, 1911. iv THE PELL EQUATION That indeterminate equation of the second degree in two unknowns which is of the form X2 - Ay2 = 1 has long been called the Pellian equation, or the Pellian problem; and it has rarely been mentioned by any other title. As will be shown, John Pell had but little to do with it; and yet, to attempt to rename it would be like trying to give another name to North America because Vespucius was not its discoverer. In referring to this as the Pellian equation, modern mathematicians have not disregarded Fermat, Wallis, Brouncker, Gauss, and the hundreds of others who have contributed to the history of the subject, just as in permitting the reciprocity theorem to bear the name of Legendre,1 the world has not failed to recognize the work of Kronecker. If the name were changed, however, then such expressions as Pellian convergents, Pellian terms, Pellian numbers, and Pellian factorizations,2 all depending upon the connection of Pell's name with the equation, would become illogical and mere arbitrary designations. The name originated in a mistaken notion of Leonard Euler3 that John Pell was the author of the solution which was really the work of Lord Brouncker. Euler in his cursory reading of Wallis's algebra must have confused 1 T. Pepin, "Etude historique sur la theorie des residues quadratiques," Memorie dell' Accademia pontificia di' Nuovi Lincei, vol. XVI, p. 229, Rome, 1900. 2 A. Cunningham, "High Pellian factorisations," Messenger of Mathematics, vol. XXXV (2), p. 166, London, 1906. (2) indicates second series. 3 P. H. Fuss, editor, "Correspondance mathematique et physique de quelques celebres geometres du XVIIIieme siecle," letter IX of L. Euler to C. Goldbach, Aug. 10, 1732, p. 37, St. Petersburg, 1843. 2 1 2 THE PELL EQUATION the contributions of Pell and Brouncker.1 Wallis2 gives Pell credit for certain researches in indeterminate analysis, but where Ay2 + 1 = x2 is discussed only Brouncker's methods are set forth. The assertion of Hankel,3 for example, that Pell treated of the equation Ay2 + 1 = x2 rests upon a misunderstanding. Hankel said, "Pell has done it no other service than to set it forth again in a much read work," i.e., in the notes to the English translation which Brancker,4 in 1668, published of the "Teutschen Algebra" of Rahn.5 Konen4 says that in the only copy which he could obtain of this work there is nothing relative to this equation, and he thinks that it is very probable that Pell never considered it. Enestr6m6 holds the same opinion, basing it upon the examination of three copies of Rahn's algebra. Nevertheless it seems not improbable that Pell solved the equation, for we find it discussed in Rahn's algebra7 under the form x = 12yy - ZZ. This shows that Pell had some acquaintance with the Pell equation, and that Euler was not so far out of the way when he attributed to Pell work upon it. 1 G. Wertheim, "Uber den Ursprung des Ausdruckes 'Pellsche Gleichung,"' Bibliotheca mathematica, vol. II (3), p. 360, Leipzig, 1901. 2 J. Wallis, "Opera mathematica," vol. II, chap. 98, p. 418, Oxford, 1693. 3 H. Hankel, "Zur Geschichte der Mathematik in Alterthum und Mittelalter," p. 203, Leipzig, 1874. 4H. Konen, "Geschichte der Gleichung t2 - Du2 = 1," Leipzig, 1901. Konen confuses Thomas Brancker and William Brouncker. 5G. Wertheim, "Die Algebra des Johann Heinrich Rahn (1659)," Bibliotheca mathematica, vol. III (3), p. 125, Leipzig, 1902. 6 G. Enestr6m, "Uber den Ursprung der Bennenung Pell'sche Gleichung," Bibliotheca mathematica, vol. III (3), p. 204, Leipzig, 1902. 7 J. H. Rahn, "An introduction to algebra, translated out of the High Dutch into English by Thomas Brancker, M.A. Much altered and augmented by D. P.," p. 143, London, 1668. THE PELL EQUATION 3 In the Pell equation, and in such generalizations as x2 -Ay2 = I 1, x2 -Ay2 = b, x2- Ay2 = a2, it has been customary to restrict A to a positive nonsquare integer, and to seek integral solutions for x and y. It is evident that no generality will be lost if the solutions are further restricted to positive integers. If it were permitted to make A < - 1, there would be only two solutions in integers, x= 1, y = 0. For A = -1, only four solutions, x= =- 1, y = O; x = 0, y=; but for A > 1, x2-Ay2= 1 has infinitely many solutions. From every two solutions, xl, yi; x2, y2, the same or dif-, ferent, a third can be obtained from the following identity, (X + Y1i A) (x2 + Y2 wA) xix2 + Ay1y2 + (Xly2 + X2y1) AA, namely, x = XlX2 + AY1y2, y = Xly2 + x2Y1. All the solutions for which x > 0, y > 0, can be obtained from the following formula in which xl, yi, is the fundamental solution, i.e., the solution in smallest integers, excepting the solution 1, 0: x + y = =(xi +y A)k (k = 1, 2, 3, * oc). It is evident that the Pell equation is closely connected with the primitive methods of approximating a square root; indeed, that the latter are, in general, special cases of the general equation. Among the first definite traces that we have of these methods of approximation are those 4 THE PELL EQUATION found in the dimensions of ancient structures, such as the Egyptian pyramids, the Parthenon and other temples on the Acropolis at Athens, and altars and temples in many places. For example, the principal room, called the King's chamber,1 in the pyramid of Cheops, has for the ratio of its height to its breadth about 1.117 or nearly 1 /5 showing that the architect must have known the approximate value of this surd; and it is suggestive, to say the least, when we note that this is one half the ratio x: y of some of the solutions of x2 - 5y2 = 1. The ratio 17: 12 is found many times on the Acropolis, and x= 17, y = 12 satisfies x2 - 2y2 = 1. These solutions are interwoven with the number theory of Pythagoras and his followers, and the ancient tradition that Pythagoras had obtained his knowledge of numbers by a sojourn upon the Euphrates is strengthened by recent discoveries.2 These solutions are found in connection with the mystic Platonic number,3 and a profound consideration of such numbers no doubt gave impulse to the researches of Theon of Smyrna in connection with his side- and diagonalnumbers. There is a very intimate mathematical relation between the Pell equation and the extraction of square roots. 1H. A. Naber, "Das Theorem des Pythagoras," p. 48, Haarlem, 1908. 2 H. V. Hilprecht, "Mathematical, meteorological and chronological tablets from the temple library at Nippur," vol. XX, part 1 of series A, cuneiform texts, University of Pennsylvania, Phila., 1906. M. Cantor, "Uber die alteste indische Mathematik," Archiv der Mathematik und Physik, vol. VIII (3), Heft 1, p. 63, Leipzig, 1904. Cantor sees no ground for supposing Pythagoras a pupil of India but rather holds Egypt to be the source of his mathematical knowledge. He refers to L. von Schroeder, "Pythagoras und die Inder," 1884. 3 S. Giinther, "Die platonische Zahl," p. 5, Dresden, 1882. THE PELL EQUATION 5 This will be noted when the problem of such extraction is not attached to the decimal system, as is the present usual custom, nor to the sexagesimal system as was the custom of the Greek astronomers. The fact that tables of squares and square roots have been discovered on tablets from the temple library at Nippur1 and from the library of Sardanapalus IV at Babylon2 has an important bearing on the methods of solution of the Pell equation. It is therefore quite possible that solutions of this equation extend back to the ancient Babylonians four thousand years ago. It will not be far out of the way to say that the first approximations to 42 appeared both in India and in Greece about four hundred years before Christ. The younger Pythagoreans (before 410 B.C.) recognized and proved the incommensurability of the diagonal and side of a square and set forth certain approximations.3 The Sulva-sutras in India, which contained approximations to I-2, are not later than the fourth or fifth century before Christ.4 The fact that these approximations appeared about the same time in Greece and India, and that the first step in these approximations appears so simp]e, indicate the independence of the Greek and Hindu mathematicians. Furthermore, the processes of root approximaHilprecht, op. cit., p. 13; D. E. Smith, "The mathematical tablets of Nippur," Bulletin of the American Mathematical Society, vol. XII (2), p. 394, New York, 1907. 2 M. L. Rodet, "Sur les methodes d' approximation chez les ancients," Bulletin de la Societe Mathematique de France, vol. VII, p. 159, Paris, 1879. 3H. Vogt, "Die Entdeckungsgeschichte des Irrationalen nach Plato und anderen Quellen des 4. Jahrhunderts," Bibliotheca mathematica, vol. X (3), p. 97, Leipzig, 1909. G. Junge, "Wann haben die Griechen das Irrationale entdeckt?" Halle, 1907. M. L. Rodet, "Sur une methode d'approximation des racines carrees, connu dans l'Inde anterieure a la conquete d'Alexandre," Bulletin de la Societe Mathematique de France, vol. VII, p. 98, Paris, 1879. 4G. Thibaut, "On the sulvasutras," Journal of the Asiatic Society of Bengal, vol. XLIV, p. 239, Calcutta, 1875. 6 THE PELL EQUATION tion were developed differently by the Hindus and the Greeks, and this adds probability to the idea of independent discovery.' Indeed, we fail to find any definite connection between the Greek and the Hindu algebra. Each starts with a simple approximation to the square root of a number and seeks from this a closer approximation. We shall see what different results are obtained. The Hindus were skilful calculators but mediocre theorists, while the genius of the Greeks tended more towards geometry. One of the monuments of the old Hindu mathematicians is that curious work, the Sulva-sutras, or "Precepts of line." These books contain rules to be observed by the Brahmins in the construction of their altars, and give close approximations for a/2. Baudhayana, the author of the oldest of these works, uses first the approximation 17/12.2 Both Baudhayana3 and Apastamba4, another of the early writers, give this rule in relation to the approximation for the diagonal of a square. "Increase the measure by its third part and this third by its own fourth less the thirty-fourth part of that fourth. The sutras are characterized by an enigmatical shortness, but the rule evidently means that 1 1 1 a2 3 +343.4 4.34' This result, 577/408, furnishes a solution of the Pell equation X2 - 2y2 = 1. 1 P. Tannery, "Sur la mesure du cercle d' Archimede," Memoires de la Societe des Sciences de Bordeaux, vol. IV (2), p. 313, Paris, 1882. S. Gunther, "Die quadratischen Irrationalitaten der Alten und deren Entwickelungsmethoden," Zeitschrift fur Mathematik und Physik, Abhandlungen zur Geschichte der Mathematik, vol. XXVII, p. 43, Leipzig, 1882. 2 M. Simon, "Geschichte der Mathematik in Altertum," p. 157, Berlin, 1909. 3 G. Thibaut, loc. cit. 4 A. Biirk, "Das Apastamba-Sulba Sfitra," Zeitschrift der Deutschen Morgenlindischen Gesellschaft, vol. LV, p. 543, Leipzig, 1901. THE PELL EQUATION 7 Perhaps some of the first approximations were found by trial with pebbles,1 taking a square number and attempting to arrange in two equal squares and repeating the process with other squares until close approximations were obtained. If 289 pebbles arranged in a square were taken up and rearranged in two squares of 144 each with 1 excess, a solution of the Pell equation 2 - 2y2 = 1 would be found in which x= 17, y= 12. Suppose p/q to be a first approximation of the square root of a number A, and that another is built up from it, thus,2 nr iNP 2T P R Pi 4A <o P- WA c P- +.- = pq q 2pq ql where p2 - Aq2 = -R, and ca is to read "approximates." This approximation is particularly close in cases where R = -+ 1 or == 2, the fraction to be added in these cases being in a form esteemed by the Greeks as especially advantageous. Where R = - 1, we have from this expression for /A a limitless series of solutions for the Pell equation x2 - Ay2 = 1. Let us examine the first3 extraction of 42 of which we have knowledge, those of the Sulva-sutras just mentioned, and see if they fit the method suggested. If now 3/2 has been found by experiment as an approximation, so 1 G. Thibaut, loc. cit. 2 P. Tannery, "Sur la mesure du cercle d'Archimede," loc. cit. 3H. G. Zeuthen, "L'oeuvre de Paul Tannery comme historien des mathematiques," Bibliotheca mathematica, vol. VI (3), p. 257, Leipzig, 1905. 8 THE PELL EQUATION that from 3 2' we have 3 -1 17 2 2+ 3.2 -12' and 17 -1 577 12 2.17 12 408' where 3, 2; 17, 12; and 577, 408 are solutions of 2- 2y2 = 1. Baudhayana1 sometimes used for l-3 the approximation 26/15, for he gave for the area of a circle the rule which which would have this meaning, (13d ) 1 d2,r Here ir is evidently set equal to 3, as in the early writings of all oriental nations. Therefore 132 3 ~ -2.4, 152 and - 26 15k Baudhayana also knew the more exact value2 2 1 1.3 3+35 3552' which value, 1351/780, was probably obtained by the 1S. Giinther, op. cit., p. 41. M. Cantor, "Vorlesungen fiber der Geschichte der Mathematik," vol. I, 2d ed., p. 548, Leipzig, 1894. 2 S. Gunther, loc. cit. THE PELL EQUATION 9 use of the formula just mentioned,1 thus: 3 26 "3 C 15' and - 26 - 1 1351 43 15 +2.26 15 780' both 26, 15, and 1351, 780, being solutions of the Pell equation 2 - 3y2 = 1. It is worth noting that the methods2 used are identical with those of finding integral solutions of the Pell equation. The simplicity of these calculations which are capable of being continued indefinitely, naturally provoked efforts to calculate other square roots. These efforts, by simple methods and by experiments natural for the calculators to make without any further arithmetic theory led up to the rules of Brahmagupta and Bhaskara. With the same starting point, WA oA p/q, the Greeks were led by somewhat different rules to closer approximations of IA, but still by methods in which the approximations satisfy the equations x2 - Ay2 = 1. Plato, who derived his number theory from the Pythagoreans, was acquainted with the approximation 7/5 for the ratio of the diagonal and side of a square,3 so that this value dates back at least to the middle of the fifth century before Christ. The Greeks considered every rational integer4 first of all as a straight line, so that 5 was 1 P. Tannery, "Sur la mesure... " loc. cit. 2 Plato, Republic (VII, 546 be.). See M. Cantor, "Vorlesungen fiber der Geschichte der Mathematik," Vol. I, 2d ed., p. 210, Leipzig, 1894. 3 G. Friedlein, "Procli Diadochi in primum Euclidis Elementorum librum commentarii," p. 427, Leipzig, 1873. 4 F. Hultsch, "Die Pythagoreischen Reihen der Seiten und Diagonalen von Quadraten und ihre Umbildung zu einer Doppelreihe ganzer Zahlen," Bibliotheca mathematica, vol. I (3), p. 8, Leipzig, 1900. 10 THE PELL EQUATION taken as a line 5 units in length and upon it a square was constructed. This square contained, accordingly, 25 square units. The diagonal of this square was /2 *25 linear units. This diagonal was called by the Pythagoreans the appr)To &dlt'CeTpos of the number 5. In conjunction with this j/50 - 1 was called the pT'} 8td/LierTpo9 which equaled 7 linear units, i.e., the rational number whose square differs from the square of the irrational diagonal by an integral minimum. And 7, 5, is a solution of x2- 2y = - 1. That the pr1T7 BL&tdCe/poq had a general significance to the Pythagoreans has become evident since the publication of Kroll'sl text of Proclus's commentary on Plato. It is based on an application of Euclid II, 10,2 which was common property before Plato and was known to the Pythagoreans as early as the middle of the fifth century before Christ. If C (Fig. 1) is the mid-point of the base A C D B FIG. 1. of an isosceles right triangle AEB, and D any point on AB, then AD2 + DB2 = 2AC2 + 2CD. For if Z is the point where the perpendicular at D meets 1Proclus, "In Platonis rempublicam commentarii," editor Kroll, vol. II, c. 27, Leipzig, 1901. 2 F. Hultsch, loc. cit. H. G. Zeuthen, "Geschichte der Mathematik im Altertum und Mittelalter," p. 59, Copenhagen, 1896. Also the French edition, H. G. Zeuthen, "Histoire des mathematiques dans l'antiquite et le moyen age," J. Mascart, ed., p. 47, Paris, 1902. M. Cantor, op. cit., vol. I, p. 249, 2d ed. THE PELL EQUATION 11 EB, DB = DZ AD2 + DZ2 = AE2 + EZ2 = 2AC2 + 2CD2. If CD = q and BD = p, AD = 2q + p = p, AC = q+ p = l, then 2q2 - p2 = (2q2 - p2). This enables us to derive from one integral solution, p, q, of one of the two equations 2 - 2y2 = l 1, a solution, pi, q1, for the other, always in larger integers. We might start from 1, 1, or from the one with which the Pythagoreans were already familiar, 7, 5. As Proclus1 pointed out, the Pythagoreans proceeded as follows:2 On AB (Fig. 2) construct a square and draw its diagonal BE. On the prolongation of AB lay off K E \ A B C D G H FIG. 2. BC = AB and CD = BE. Then according to the Pythagorean theorem CD2 = 2AB2, and by the theorem just referred to AD2 + CD2 = 2AB2 + 2BD2 1 Proclus, loc. cit. 2 Hultsch, loc. cit. 12 THE PELL EQUATION Consequently, after subtraction, AD2 = 2BD2. Now erect on BD a square and draw its diagonal. Then DF2 = 2BD2. Therefore DF = AD = 2A CD 2A BCD AB + BE. Upon the prolongation of AD, lay off DG = BD, and GH = DF; and on DH erect a square and draw its diagonal HK. Then, by similar steps, HK = BH = 2BD + GH = 2BD + DF. It is clear that, in the continuation of this construction, in each new square the diagonal equals the sum of the diagonal and double the side of the previous square. If we designate the sides of the successive squares by s1, S2, S3,..., and the corresponding diagonals by di, d2, d3, * * *, we have the double series, Si, 2 = s + d1, s3 = S2 + 2,... dl, d2 = 2sl + d, d3 = 2s2 + d2,... in which the double square of each term in the upper line is equal to the square of the corresponding term in the lower line. This geometric proof was, however, for the Pythagoreans the introduction to some fine considerations in the theory of numbers. If they set s = 1, then d = /2 and this was, like 1/50 just referred to, an "unspeakable" number; and in connection with this comes into view the p'wnr 8tdaieIpoS, the nearest integer, the number 1. Then they compared the square of the dpprT7o and the p7 TB lCtd-epos, the first 2, the second 1, the difference 1. If we now distinguish the successive p3rTa Stdl/eTpot as 56, 62, 33,..., and substitute them in the double series THE PELL EQUATION 13 they change to s1 =1, S2 = s1 + 1=2, 83 = S2 + 62 = 5,.. 51 = 1, 52 = 2si + 61 = 3, ~3 = 2S2 + 82 = 7, *. Then no longer, as in the first double series, does the square of each lower member equal double the square of the upper, but the difference is constantly 1 and alternately + and -. Each 5r, Sr, is a solution of one of the Pell equations 2 - 2s2 = _ 1. Theon1 of Smyrna, who lived about 130 A.D., set forth this development earlier than Proclus. He called these numbers side- and diagonal numbers, and both he and Proclus carried the computation as far as 2.122 + 1 = 172. Theon evidently understood the generality of the double series for he said, "In this manner the diagonals and sides will always be expressible," perhaps'omitting the general proof2 because it was so evident. Expressed in algebraic symbols, if X2 - 2y2 = _ 1, suppose x = p, y = q and put xl = p + 2q, yl = p + q; then X12 2y12 = p2 + 4pq + 4q 2- 2p2 - 4pq - 2q2 = - p2 + 2q2 = - (p2 - 2q2). But evidently x"=y=l is a first solution of x2 - 2y2 - 1. Therefore X12-2y~2 = =F 1. Theonis Smyrnaei, "Platonici, eorum, quae in mathematicis ad Platonis lectionem utilia sunt, expositio," p. 67, Lutetiae Parisiorum, 1544. "Oeuvres de Theon de Smyrne," J. Dupuis, editor, p. 71, Paris, 1892. 2G. H. F. Nesselmann, "Die Algebra der Griechen," p. 229, Berlin, 1842. Compare M. Cantor, vol. I, p. 408, 2d ed. 14 THE PELL EQUATION Theon adds by way of conclusion, and this is also pointed out by Proclus, that the sum of the squares of all the fprval 8dfaeTpot will be twice as large as the sum of the squares of all the sides. If this sum is limited to a finite number of terms of the double series, then the number of terms must be even. Hultschl takes the view that Proclus did his work independently of Theon, but that both spring from a common origin, for he says, "If Proclus, as many signs indicate, has drawn from the tua67 ldT7rwv e(wDpi'a of Geminos, so Theon has done so also, not directly but through a second or third hand." Guinther2 gives a different explanation whereby Theon might have arrived at his numbers. From y + P (x + y), y - q (x - y), q p it follows, if p2 + 2pq - q2 = z2, 4q2 + z = 2q 2q 42 z q2 - z 4q 12q2 z y=, x=, z z and when z = 1, any number q is found for which 2q2 + z is a perfect square, this solution gives an infinite number of integral values of y and x; and the solution of the general equation (a2 b2)y2 - = X2 is accomplished. Besides the geometric motive which in the last half of the fourth century before Christ, led to the discovery of approximations for J2, and for such numbers as those of Theon at about the same time, the problems of music also led to new researches to determine as exactly as possible J12. The Pythagorean concept of the harmonic Hultsch, loc. cit. 2S. Gunther, "Uber einen Specialfall der Pell'schen Gleichung," Blatter fur das bairische Gymnasial- und Realschulwesen, vol. XVIII, p. 19, Munich, 1882. THE PELL EQUATION 15 mean, as its name implies, is derived from a problem of music. Tannery1 says of the numbers of Theon, "I conceive this series as obtained upon the numbers themselves, without any immediate generalization, without any foresight, and without perhaps any demonstration of this characteristic property, but I believe there is a simpler means of explaining their invention, and this means is intimately connected with the consideration of the harmony of Pythagoras which gives the degree of approximation p/q = 3/2, 2q/p = 4/3." As the solutions of X2 - 2y2 = = 1 were evidently made in order to obtain approximations to the ~/2, so when we find Archimedes" giving without explanation the fractions 265/153 and 1351/780 as approximations to a/3, the most natural hypothesis is that he obtained them from similar equations with 3 substituted for 2. Archimedes was the first Greek mathematician who was not content to speculate over irrationals but handled them in computation. He expressed the sides of regular inscribed and circumscribed polygons in terms of the radius of the circle. In particular for a regular hexagon he gave the proportion r: 3 *2 and then set 1351 265 780 153' There has been much speculation as to how these approximations were obtained.3 Let us look at one of the solu1 P. Tannery, "Du r6le de la musique grecque dans le developpement de la mathematique pure," Bibliotheca mathematica, vol. III (3), pt. 161, Leipzig, 1902. 2 T. L. Heath, "Works of Archimedes," p. LXXX, Cambridge, 1897. 3 P. Tannery, "Sur la mesure du cercle d'Archimede," Memoires de la Societe des sciences de Bordeaux, vol. IV (2), p. 303, Paris, 1882. H. G. Zeuthen, "Nogle hypotheser on Arkhimedes kvadratsrodsheregning," Tidsskrift for Mathematik, vol. III (4), p. 145, Copenhagen, 1879. 16 THE PELL EQUATION tions suggested by Tannery' with the same procedure as that hinted at on the preceding page for /2. If p/q is an approximation for the -13, then 3q/p is; and we take pi = p + 3q, qp = p + q. But here Pi2 - 3q22 = - 2(p2 - 3q2) and if we start from the difference + 1 (as for p = 2 and q = 1), the following difference will be - 2, and the next + 4. But the terms of the corresponding relations are even numbers, as it is easy to see. Therefore if we take 2p2 = p - 3ql, 2q2 = pi + ql, we find + 1 again for a difference, thus: 22 - 3q22 = 1. We obtain then a complete series of solutions as follows, writing the solutions for x2 - 3y2 = and x2 - 3y2 = - 2 in separate rows, 2 7 26 97 362 1351 1' 4' 15' 56' 209' 780' * pi = 2p +3q, 5 19 71 265 989 q, = p + 2q, 3' 11' 41' 153' 571' * and from these double series Archimedes could choose the terms which were convenient for the degree of approximation that he desired. Zeuthen seems to have been the first to connect the ancient approximations of /3 with the Pell equation. He and De Lagny2 (1723), give methods somewhat similar to Tannery's. 1 P Tannery, "Du role de la musique," Bibliotheca mathematica, vol. III (3), p. 174, Leipzig, 1902. 2 S. Gfinther, "Antike Naherungsmethoden im Lichte moderner Mathematik," p. 16, Prague, 1878. (Reprint from Abhandlungen der koniglichen bohmischen Gesellschaft der Wissenschaft, vol. IX (6), Mathematik.Naturwiss. Classe, No. 4.) THE PELL EQUATION 17 Heilermann's' method has an interest in that he brings out the two Archimedian approximations in immediate sequence without intervening values. He generalizes the side- and diagonal numbers of Theon of Smyrna, which furnished solutions for the equations X2 - 2y2 = - 1, into solutions of the general Pell equation X2 - ay2 = b. From Theon we might have Sn = Sn-1 + D_, Dn n = 2Sn- + Dn-1. Heilermann set S1 = So + Do) D1 = aSo + Do, S2 = Si + D1, D2 = aSD + D1, S3 = S2 +, D3 = aS2 + D2, *.................. Sn = Sn-_ + Dn-1, Dn = aSn-1 + Dn1. Therefore aS,2 = aS,-_2 + 2aSn_lDn_l + aDn-2, Dn2 = a2Sn-2 + 2aSn-_Dn-_ + Dn1_2. Subtracting, Dn2 - aSn2 = (1 - a)(D,_2 - aSn_2) = (1 - a)2(Dn_22 - aS_22) = (1 - a)n(D2- aS2); and, if Do = So = 1, Dn2 - aSn2 = (1 -a),+l; and since D,n + (1 - a)"n+ S n- (1 S-2 1 Heilermann, "Bemerkungen zu den Archimedischen Naherungswerten der irrationalen Quadratwurzeln," Zeitschrift fiir Mathematik und Physik, Historisch-literarische Abtheilung, vol. XXVI, p. 121, Leipzig, 1881. 3 18 THE PELL EQUATION this is clearly an approximation for ala. If we set b2a in place of a we may thus make a new a that is somewhat nearer to unity. Thus suppose we have 27 a 25' so that 3 5 and put Do = So = 1; then 52 5 26 26 S = 2, D3 or 1=2, =251' 4 3 25 r15' 102 54 + 52 - 5 106 265 S2 - 25' 25D2- 3 1- - or S =25 25 ' 3 3 102 153' 208 5404 5 5404 1351 25' D325 25' 3 25 208 r 780; the two Archimedian approximations coming close together. Other methods are given by Hunrath, Hultsch, Rodet, Oppermann, Sch6nborn, many of the solutions depending on the relation b b a 2a > a2 + b > a i 2a -- 1 where a2 + b is a non-square integer and a2 is the nearest square number to it. If in the general equation X2- ay2 = b we have found by trial the three simplest solutions, Tannery has shown the law of forming successive solutions.' If x=p, y =q 1 P. Tannery, "Sur la mesure...," loc. cit. T. L. Heath, " Diophantus of Alexandria," p. 279, 2d ed., Cambridge, 1910. THE PELL EQUATION 19 is a known solution, put pi = ap + -q, q1 = yp + q, and it is sufficient to know the three simplest solutions in order to find a, 3, y, 5; for, substituting the values of p, q, Pi, ql, p2, q2, where p2, q2, are formed from pi, qi, by the same law as pi, q1, are derived from p, q, we have four simultaneous equations in four unknowns. Taking the particular equation x2 - 3y2 = 1 we easily find the first three solutions 1, 0; 2, 1; 7, 4. Whence 2 = a, 1 = y, 7 = 2a +, 4 = 2y + 5, and a = 2, Y = 3, = 1, 6 = 2, so that pi =2p + 3q, q = p + 2q, as already stated. The celebrated Cattle Problem of Archimedes must also be mentioned. Tannery1 says that instead of the name of Pell in connection with the equation x2 - Ay2 = 1, the name of Archimedes ought to be joined to that of Fermat, for the Cattle Problem shows that Archimedes conceived the problem of the Pell equation in all its generality. In this problem it is required to find the number of bulls and cows of each of four colors, or to find eight unknowns. The problem has a natural conclusion at the close of seven conditions. Then, evidently by a later hand, two other conditions are added; and the language of the eighth is so ambiguous that it is doubtful whether the sum of the number of bulls of two colors is a square number or merely a number with two factors nearly equal. But if all nine conditions are taken and the eighth is made to refer to a square number, then the 1 P. Tannery, "Du role de la musique...," p. 175. 20 THE PELL EQUATION Cattle Problem involves the solution of the Pell equation x2 4,729,494y2 =. But it is very doubtful whether the Cattle Problem was proposed by Archimedes or even in his time.1 If he did propose it he certainly could not solve it.2 Nesselmann3 sums up the evidence by saying that the problem is clearly at an end after seven conditions have been given, the concluding words being "he who solves the problem must not be unskilled in numbers." Its language and versification are against its authenticity. The scholiasts' solution does not, as it claims, satisfy the whole problem but only the first part. The impossibility of solution with Greek numerals and the very large numbers used show that the author, or authors, could not have seen what the effect of the many heterogeneous conditions would be. There is nothing impossible in the supposition that Archimedes was in the possession of a general method of solving such equations when the numbers involved were not too great for manipulation in the Greek numerical notation.4 But the other conditions are such that we are not allowed to take the fundamental solution of the Pell equation but must take a solution in which y is divisible by 9,314, with the result that larger numbers of cattle are involved than could by any possibility stand upon the plains of Thranika or the whole island of Sicily for that matter, numbers beyond comprehension or possibility of computation. For instance the number of yellow bulls would be 639,034,648,230,902,865,008,559,676,183....... 635,296,026,300, where the dots indicate the omission of 68,834 periods of 1J. Struve and K. L. Struve, "Altes griechisches Epigramm," Altona, 1821. 2 S. Gunther, loc. cit. 3 G. H. F. Nesselmann, loc. cit. 4T. L. Heath, "Diophantus of Alexandria," loc. cit. THE PELL EQUATION 21 3 figures each; and though Archimedes had machinery for dealing with very large numbers, he had not such as would justify the assumption that he could solve this problem. The fact that he could not solve it would not, however, preclude his proposing it to the Alexandrian mathematicians.1 The above number is taken from the results of A. H. Bell,2 whose solution, derived from the equation 2 - 410,286,423,278,424y2 = 1, is the most complete that has appeared. An exhaustive literary discussion of this problem is given by Dr. B. Krumbiegel.3 There is also a very full solution by Dr. A. Amthor4 in which the equation x2 - 4,729,494 y2 = 1 is solved by reducing 44,729,494 to a continued fraction with a period of 91 terms, and to which various corollaries are appended showing how to find the smallest value of y which shall be divisible by 9,314. Heron5 of Alexandria is noted for his approximation to a/3. He used 26/15 in connection with finding the area of an equilateral triangle. He found approximations to many other surds, giving results which correspond exactly to solutions of the Pell equation. The form in which Heron expresses the approximations is suggestive of Egyptian methods. The following are a few of his 1 According to Struve this problem was discovered in modern times by Lessing, "Beitrage zur Geschichte und Literatur," p. 423 (1773). See also J. Gow, "History of Greek mathematics," p. 99, Cambridge, 1884. M. Merriman, "Cattle Problem of Archimedes," Popular Science Monthly, vol. LXVII, p. 660, New York, 1905. 2 A. H. Bell, "On the Cattle Problem of Archimedes," Mathematical Magazine, vol. II, p. 163, Washington, 1895. 3 B. Krumbiegel, "Das Problema bovinum des Archimedes," Zeitschrift fur Mathematik und Physik, Historisch-literarische Abtheilung, vol. XXV, p. 121, Leipzig, 1880. 4 A. Amthor, same title and journal as the preceding reference, p. 151 5 F. Hultsch, editor, "Heronis Alexandrini geometricorum et stereometricorum reliquae," Berlin, 1864. 22 THE PELL EQUATION approximations,l with their connection indicated: Surd. Heron's approximation. 450 7 +, 1 1 1 4135 l 11 — + - 2 14 21' 2 1 /1575 ~ 39 + 2 + 2 1 216 c 14 ++ 33 1 1 720 c\ 26 ++, 2 3 1 Equivalent fraction. 99 14' 244 21' 2024 51 ' 485 33 ' 161 6 ' Corresponding to the fundamental solution of X2 - 50y2 = 1; x2 - 135y2 = 1; x2 - 1575y2 = 1; x2 - 216y2 =1; X2- 720y2 = 1. Diophantus flourished about 250 A.D., or not much later. In discussing his relation to the problem, we must remember that he did not avoid fractional solutions to indeterminate equations but sought merely rational solutions. But there are many cases in which he actually does find integral solutions. In Book V of his Arithmetica,3 problem 9, we have the equation whence Therefore 26y2 + 1 = x2; 26y2 + 1 = (5y + 1)2, say. y= 10, where the corresponding value of x would be 51. In Book V, 11, we have the equation whence 30y2 + 1 = x2; 30y2 + 1 = (5y + 1)2, say. 1 S. Giinther, loc. cit. 2 This is as old as Pythagoras. T. L. Heath, "The works of Archimedes," p. lxxix, Cambridge, 1897. 3 T. L. Heath, "Diophantus of Alexandria, a study in the history of Greek algebra," 2d ed., p. 207, Cambridge, 1910. THE PELL EQUATI()ON 23 Therefore y=2 and x= 11. The equation1 Ay2 + C = x2 is handled by Diophantus as follows. It can be rationally solved if (1) A is a positive square, say a2. Thus a2y2 + C = X2 In this case x2 is put equal to (ay i m)2. Therefore a2y2 + C = (ay n m)2, and C - m2 y = ~ d 2ma (m and the doubtful sign being always assumed so as to give x a positive value.) When (2) C is positive and a square number, say c2, Ay2 + c2 = 2. Here we may let x = my M = c. Therefore Ay2 + c2 = (my = c)2, and 2mc y= A Diophantus shows how when one solution to the equation is known any number of others may be found. The lemma to VI, 15,2 reads: "Given two numbers, if, when some square is multiplied into one of the numbers and the other number is subtracted from the product, the result is a square, another square larger than the 1G. H. F. Nesselmann, "Die Algebra der Griechen," p. 329, Berlin, 1842. 2 T. L. Heath, "Diophantus," p. 238. 24 THE PELL EQUATION aforesaid square can always be found which has the same property." It would not have been a long step for Diophantus to have generalized this to the case - Ay2 =- C = x2 and then to the form Ay2 + By + C = x2. Suppose that one solution, p, q, of Ay2 - C = x2 has been found. Diophantus then proceeds to substitute in the original expression y = q +m, x= p - km, where k is some integer. Then A(q + m)2- C = (p - km)2. Since Aq2 - C = p2, we have 2m(Aq + pk) = m2(k2 - A), whence 2(Aq + pk) m ~k2 -A and 2(Aq + pk) y = k2 -A Diophantus also notes a necessary condition for x2 - Ay2 = _ c for he says in VI, 14, that the equation can not be solved unless A is the sum of two squares. Tannery' thinks that Diophantus may have discussed the equation x2 - Ay2 = 1 1 P. Tannery, "L'arithmetique des Grecs dans Pappus," Memoires de la Societe des sciences de Bordeaux, vol. III (2), p. 370, Paris, 1881. THE PELL EQUATION 25 more fully in that part of the Arithmetica which is lost. This is his suggestion of how Diophantus might have proceeded to find a more general solution, Pi, q1, from a given solution p, q, of the equation x2 - Ay2 = 1. Let Pi = x - p, and q = x + q; then pi2 - Aq12 = m2x2- 2mpx + p2 Ax2 - 2Aqx - Aq2 = 1. Then since p2 - Aq2 = 1, mp + Aq m2 -A,' (m2 + A)p + 2Amq 2mp + (m2 + A)q pi = m2 -A 1 m2 - A, and in fact p-2 - Aq12 = 1. If an integral solution is wanted it can be had by substituting u/v for m where u, v, is a solution of the equation u2 - Av2 = 1; i.e., another solution of the original equation. Then pi = (u2 + Av2)p - 2Auvq, q, = 2puv + (u2 + Av2)p. In order to have the above values for pi, qi, correspond with the simpler values given on page 16 for the solution of the equation x2 - 3y2 = 1, it is necessary to take u, v, not a solution of the equation x2 - 3y2 = 1 but as the simplest solution of the equation x2 -3y2 = -2. 26 THE PELL EQUATION Then when u= 1, v= 1, satisfy the equation x2 _ 3y2 = 2, we have (u2 + 3v2)p - 6uvq 4p + 6q Pi = u2v2 -2 - (2p + 3q), and 2uvp + (u2 + 3v2)q 2p +- 4q q = 32 = - (p + 2q), and the positive values for pi, ql, could also be taken since they satisfy the equation pl2 - 3ql2 = 1. Thus in order to get a general solution, Diophantus required two known solutions of the original equation, or one of the original and one of an auxiliary equation. Although the solution of the Pell equation is not explicitly found in the writings of Aryabhatta (c. 525 A.D.), it is by no means certain that he knew nothing of it,l for he recorded only so much algebra as he conceived to be necessary for his astronomy. Moreover, Aryabhatta had a definite idea of the approximation formula aa2 + r a + 2 which would lead to solutions of the equation, for he says in one of his rules:2 "Square having been subtracted from square always the non-square must be divided by double the square root. The quotient in a place set apart is the root." Turning to the works of Brahmagupta (c. 650 A.D.), we find a wealth of material bearing directly on the equa1 H. Hankel, p. 203. 2 G. R. Kaye, " Notes on Indian mathematics," Journal and Proceedings of the Asiatic Society of Bengal, vol. IV, No. 3, p. 119, Calcutta, 1908. At the conclusion of his article the author gives a bibliography of Indian mathematics. THE PELL EQUATION 27 tion. In the eighteenth chapter of his algebra, which he called Brahme-sphuta-sidd'hanta, we have the following "Rule:l A root [is set down] two fold: and [another deduced] from the assumed square multiplied by the multiplier, and increased or diminished by a quantity assumed. The product of the first [pair] taken into the multiplier, with the product of the last added, is a 'last' root. The sum of the products of oblique multiplication is a 'first' root. The additive is the product of the like additive or subtractive quantities. The roots [so found], divided by the [original] additive or subtractive quantity, are [roots answering] for additive unity." That this rule applies to the solution of the Pell equation will be more evident by the statement of the problem following the rule. "Question 27. Making the square of the residue of signs and minutes on a Wednesday, multiplied by ninetytwo... with one added to the product [afford]... an exact square, [a person solving this problem] within a year [is] a mathematician." In modern language solve the indeterminate equation X2 - 92y2 = 1. Brahmagupta's procedure is evident from a careful study of the above rule. In simplifying it we let L stand for "least," which corresponds to y in the equation and to "first" in the rule, and we let G stand for "greatest," which corresponds to x in the equation and to "last" in the rule. Brahmagupta assumes that the square is 1, its root is "least" root and he sets it down twice, thus: L 1, L 1. He multiplies this square by 92 and adds 8 to make it 1 H. T. Colebrooke, "Algebra, with arithmetic and mensuration from the Sanscrit of Brahmegupta and Bhaskara, " p. 363, London, 1817. 28 THE PELL EQUATION yield a square, the root of which, 10, he calls "greatest" and sets down the result as follows: L 1, G 10, A 8, L 1, G 10, A 8, where A 8 denotes the number which must be added to 92L2 to make the sum a perfect square. Then by the rule, 12.92 + 102 makes a new G, and 1.10 + 1-10 makes a new L, where the new number to be added is 8.8, giving L 20, G 192, A 64. By the last part of the rule, dividing by 8, we obtain, 5 &L L2, G 3 A 1. If L and G are divided by 8, A is divided by 8 8. Repeating the rule on 5 L, - <G % Al, 1, L2, G A1, we get L 120, G 1151, A 1, whence y = 120, x= 1,151 is the fundamental solution of x2 92y2 = 1. The next rule shows that the Hindus knew how from one solution to obtain others. "Rule: Putting severally for additive unity under roots for the given additive or subtractive, the 'last' and 'first' roots [thence deduced by composition] serve for the given additive or subtractive." Example. L 30, G 60, A 900, L 1, G 2, A 1, THE PELL EQUATION 29 from which we get, since 30.2 + 1.60 = 120 and 3.1-30 + 60.2 = 210, L 120, G 210, A 900. The next rule is one which saves much labor in the computation of the solutions of many of the equations, but it does not produce the fundamental solution, nor does it give integral values unless y is even. "Rule: When the additive is four, the square of the last root, less three, being halved and multiplied by the last, is a last root; and the square of the last root, less one, being divided by two and multiplied by the first, is a first root of additive unity." If X12 - Ay2 = 4, then (12 - 3) y(12- 1) 2 ' 2 are roots of the equation 2 - Ay2 = 1. From the rule for "subtractive four," if x12 - Ay12 - 4, we find in a similar manner that y = (X + 3) X12 + are roots of the equation x2 - Ay2 = - 1. These rules of Brahmagupta and those of Bhaskara 30 THE PELL EQUATION which lead to the integral solutions of x2 - Ay2 = =1 1 are the most important developments of ancient Hindu mathematics. We can hardly suppose that they originated with Brahmagupta. The very wording of the problem we have quoted would imply that such problems were to some extent known and that he expected his pupils to be successful in their solution. Sometimes he does not understand the rules he gives, and certain of the rules are followed by inappropriate examples.1 In several cases he solves part of an example and says, "The purport of the rest of the question is shown further on." Again he finds fault with a rule and says, "What occasion then is there for it?"2 Many other passages3 could be cited which go to show that he is not the author of the rules which he gives. Some think that these facts point to a dependence4 of the Hindus on Greek algebra, but in view of the early Hindu extraction of square root and its natural consequences, which have already been dis1 H. T. Colebrooke, p. 366. 2 H. T. Colebrooke, ~ 64. 3 G. R. Kaye, "Sources of Hindu mathematics," Journal of the Royal Asiatic Society, London, July, 1910. H. G. Zeuthen, "Histoire des mathematiques," p. 239. 4 G. R. Kaye, "Notes on Indian mathematics-Arithmetical notation," Journal and Proceedings of the Asiatic Society of Bengal, vol. III, No. 7, p. 501, Calcutta, 1907. Kaye says: "The only resemblance between the matter of the Bakhshali manuscript and Brahmagupta's work that Dr. Hoernle points out lies in the fiftieth sutra of the MS. and chapter XVIII, ~ 84 of Brahmagupta's algebra. Peculiar significance attaches to this problem, for it was fully dealt with by Diophantus and fully expounded in the algebra of Alkarkhi which was based on that of Diophantus. The problem in the Bakhshali MS. expressed in modern notation is 2 + 5 = m2, x2- 7 = n2, and is based on the fact that 1/4((5 + 7)/2 + 2)2 + (5 + 7) is a perfect square. This formula is given by Alkarkht. This rather remarkable coincidence unmistakably points to Diophantus as one of the ultimate sources of both Brahmagupta's work and the Bakhshali arithmetic." THE PELL EQUATION 31 cussed, we must regard the case of those who would show the dependence of the Hindus on outside sources as not made out. Dr. Hoernle1 says that Indian arithmetic and algebra, at least, are entirely of native origin. Strachey2 speaks of the superiority of the Hindus over the Greeks as being conspicuous in the excellence of their method. He says, "To maintain that the Bija Ganita rules for the solution of indeterminate problems might have been had from any Greek or Arabian, or any modern European writer [before Bachet de Mezeriac, Fermat, Euler or Lagrange] would be as absurd as to say that the Newtonian astronomy might have existed in the time of Ptolemy." He emphasizes the dissimilarity between the methods of Diophantus and of the Hindus.3 In addition to the rules of Brahmagupta, Bhaskara gives some alternative methods, particularly the "cyclical method," which, as Hankel4 has pointed out, has considerable significance as looking forward to the development of the modern theory of quadratic forms. The rules of Bhaskara5 which apply to our equation begin as follows: "Rules for investigating the square root of a quantity with additive unity: Let the number be assumed, and be termed the 'least root.' That number, which, added to, or subtracted from, the product of its square by the given coefficient, makes the sum (or difference) give a squareroot, mathematicians denominate a positive or negative additive; and they call that root the 'greatest' one. "Having set down the 'least' and 'greatest' roots and the additive, and having placed under them the same or others, in the same order, many roots are to be deduced 1 Hoernle, "Indian antiquities," vol. XVII, p. 37, as quoted by G. R. Kaye, loc. cit. 2E. Strachey, "Bija Ganita; or the algebra of the Hindus," p. 7, London, 1813. 3 Edinburgh review, vol. XXI, p. 372, Edinburgh, 1813. 4 H. Hankel, p. 200. 5 H. T. Colebrooke, p. 170. 32 THE PELL EQUATION from them by composition. Wherefore their composition is propounded. "The 'greatest' and 'least' roots are to be reciprocally multiplied crosswise; and the sum of the products to be taken for a 'least' root. The product of the two (original) 'least' roots being multiplied by the given coefficient, and the product of the 'greatest' roots being added thereto, the sum is the corresponding greatest root; and the product of the additives will be the [new] additive. "Or the difference of the products of the multiplication crosswise of the greatest and least roots may be taken for a 'least' root; and the difference between the product of the two [original] least roots multiplied together and taken into the coefficient, and the product of the greatest roots multiplied together, will be the corresponding 'greatest' root: and here also the additive will be the product of the two [original] additives. "Let the additive divided by the square of an assumed number be a new additive; and the roots, divided by that assumed number, will be the corresponding roots. Or the additive multiplied [by the square], the roots must, in like manner, be multiplied [by the number put]. "Or divide the double of an assumed number by the difference between the square of that assumed number and the given coefficient; and let the quotient be taken for the 'least' root, when one is the additive quantity; and from that find the 'greatest' root."' The commentator Crishna-bhatta has demonstrated these rules in a cumbersome manner in which he has used for symbols the first syllables of some of the words involved. Colebrooke has abbreviated this, and in 1Lord Brouncker's first solution, 2r 2 r2 + A )2 A ~r~A) +-l=(r-7-__ shows a marked resemblance to this rule. This was before he began to seek for integral solutions. THE PELL EQUATION 33 modern symbols the proof is as follows: If aq2 + s = p, and aql2 + s1 = pi2, multiplying both members of the first equation by pl2, and substituting aq12 + sl for its equal, p12, when it is multiplied into s, we have aq2p12 + saql2 + SS1 = p2l2. In the second term of the left member substituting for s its equal p2 - aq,2 we have aq2p12 + aq2p - a2q2q12 + ss =p p2p2. Transposing the negative term, and adding == 2appiqqi to both sides, we have aq2p12 ' 2applqql + aq12p2 + ssi = p2pl2 = 2applqqi + a2q2q22, and a(qpl = qip)2 + SSI = (ppi — 1 aqql)2. Then x = ppi = aqql, y = qpl I qlp are solutions of the equation ay2 + ss1 = X2. By repeating these steps and by applying the last part of the rule ssl becomes equal to 1; that is, ay2 + 1 = X2 is solved. Then, as Bhaskara remarks, "by virtue of [a variety of] assumptions and by composition either by sum or difference, an infinity of roots may be found." Let us examine "the splendor point of the collected wisdom of the Hindus," the method of solution which 4 34 THE PELL EQUATION they called "cyclic." The commentator Surin explaining the name cyclic uses these words, "Finding from the roots a multiplier and quotient; and thence new roots; whence again a multiplier and a quotient, and roots from them; and on in a continued round." "Rule' for the cyclic method: Making the 'least' and 'greatest' roots and additive, a dividend additive and divisor, let the multiplier be thence found. The square of that multiplier being subtracted from the given coefficient, or this coefficient being subtracted from that square (so as the remainder be small); the remainder, divided by the original additive, is a new additive; which is reversed if the subtraction be [of the square] from the coefficient. The quotient corresponding to the multiplier [and found with it] will be the 'least' root: whence the ' greatest' root may be deduced. With these the operation is repeated, setting aside the former roots and additive. This mathematicians call that of the circle. Thus are integral roots found with four, two or one [or other number] for additive: and composition serves to deduce roots for additive unity, from those which answer to the additives four and two [or other number]." The words "or other number" were evidently inserted in the rule by some commentator or by the translator and might well be omitted. The cyclic method is to be continued until the right member of x2 - Ay2 = s becomes - 4 or - 2 or - 1, and then a single step which Bhaskara calls composition brings us to the solution of the equation x2 - Ay2 = 1. The steps in the cyclic method of solution of the equation (1) ay2 + 1 = x2 1H. T. Colebrooke, p. 175. THE PELL EQUATION 35 are as follows: Choose an equation (2) aq2 + s = p2 where p, q, are relatively prime, and where to abbreviate computation, s is made as small as possible, although this latter assumption is not essential. Then p, q, s, are known. Two integers, q1 and ri, are now determined such that p +q r1 (3) q 1 which can be done by selecting an appropriate value for rl. For this purpose the Hindus used a process called pulverizer, analogous to that of continued fractions. At the same time r12 - a is made as small as possible. Then the expresssion ri2 - a 81 - is an integer.1 From this we obtain aql2 + S1 = p2. 1To prove that (r12 - a)/s is always an integer substitute the value of rl from the equation, p + qrl thus: 2q12 - 2sqlp + p2 _ r2 - a _-q2_ s2q,2 - 2sqlp + p2 - q2a. s s sq2 Substituting for p2 - q2a its value s, from the second equation, we have r2 - a s2q2 - 2sqlp+ s sq12-2qp +1 sql -p2q - 2 + p2ql2 - 2qlp + 1 S sq2 q2 q2 Substituting for s - p2 its value - aq2, from (2), we have rl2 -a -aq2q2 + (pql - 1)2 _aq2 + (p — 12 s q q Now combining (2) and (3) to eliminate s, and qip2 - aq2q1 = p + qrl, and p(pql - 1) = q(rl + aqqi); 36 THE PELL EQUATION We apply the same series of operations to the equation just written that were applied to the first equation, and obtain aq22 + s2 = P22, and so on repeatedly until Sr = 1, - 2, or 4. If one solution of aq2 + s = p2 is known, then a solution of ay2 + s2 = x2 is X = p2 + aq2, y = 2pq; and if s = - 2, 2pq is even, and therefore x2 = ay2 + 4 is divisible by 4, so that (p2 + aq2) and pq form an integral solution of ay2 + 1 = 2. One way of handling the case s= 4 has been mentioned on page 29. The Hindus applied the cyclic method successfully to all problems of this kind which they attempted, even to those involving numbers of considerable magnitude. For x2 - 61y2 = 1, Bhaskara finds y = 226,135,980, x = 1,766,319,049. He does not avoid negative numbers but uses positive or and since p and q are relatively prime (pq- - 1) is divisible by q, and therefore, (pql - l)/q is an integer, pi; and from above, (r12 - a)/s is an integer, sl, and the sum aq12 + sl is the square of an integer. THE PELL EQUATION 37 negative right members according to convenience. The following is Bhaskara's solution of the problem just mentioned, taken from Strachey's1 translation of a Persian manuscript of 1634, which is considerably clearer than the wording of Colebrooke. "What square is that which being multiplied by 61, and the product increased by 1, will be a square? Let 1 be the less root; 8 is the greater; and 3 the augment affirmative. Applying the operation of the multiplicand it is thus: Dividend Divisor Augment 1 3 8 Reject the divisor twice from the augment, 2 remains; and after the operation 2 the multiplicand, and cipher the quotient are obtained. As the line is odd we subtract cipher from the dividend and 2 from the divisor. It is 1 and 1. As we reject the divisor twice from the augment, we add 2 to the quotient. The quotient is 3 and the multiplicand 1. If we subtract the square of the multiplicand, which is 1, from 61, a greater number remains. We therefore add twice the dividend and the divisor to the quotient and the multiplicand. The quotient is 5 and the multiplicand 7." All these words are used to explain how the value of rl in (3) is obtained. The solution continues: "Subtract the square of 7 from 61; 12 remains. Divide by the augment of the operation of multiplication of the square which is 3 affirmative; 4 affirmative is the quotient; and after revision it is 4 negative; and this the augment; and the quotient which was 5 is the less root; 39 then will be the greater root. As 4 is not the original augment we have found 2 an assumed number, and by its square we divided this augment. 1 the augment negative is the quotient. We also divide 5 and 39 by 2. These same two numbers, with the denominator 2, are the quotients. E. Strachey, p. 46. 38 THE PELL EQUATION As our question is of the augment affirmative perform the operation of cross multiplication. When we multiply the augment negative by itself it will be affirmative.... The less root is 195 second parts; the greater 1,523 second parts; and the augment 1 affirmative...." The roots are "multiplied crosswise" twice and the correct roots mentioned in the beginning are obtained. The solution closes with the words: "And in like manner wherever the augment is required negative, we must multiply crosswise two augments of different sorts; and if affirmative two of the same sort." Bhaskara's next rule shows that he appreciates a necessary condition1 for the solution of x2 - Ay2 = - 1. "If the multiplicand of the question is the sum of two squares, and the augment 1 negative; it may be solved by the foregoing rules, and if wished for it may be done in another way.. " Another'rule reads: "If the multiplicand is such that you can divide it by a square without remainder, divide it; and divide the less and greater roots by the root of that square, another number will be found. And if you multiply it by a square and multiply the less and greater by its root, the numbers required will be found." Bhaskara calls that part of his work in which these rules are found the Introduction: "And this which has been written is the introduction to Indian algebra." Farther on under the head of indeterminate problems the rules are referred to and applied in many cases. These problems show also that the general form ax2 + bx + c = y2 could be transformed into the Pell equation by the given rules.2 1 This is a necessary but not a sufficient condition. 2 E. Strachey, p. 79, 80, 81. H. T. Colebrooke, p. 245, 267. THE PELL EQUATION 39 Strachey tells of finding numerous calculations which are full of errors,l showing that the operation was not well understood. Nevertheless it is evident that the Hindus knew the problem, and they seriously attempted its solution. Turning from the Hindu to the Arab civilization we find that the problem plays a still more important part in the mathematical interests of scholars. The Arabs seem to have taken their algebra largely from Diophantus, and like him they apparently made no effort to solve the Pell equation in integers. The Arithmetica of Diophantus was translated into Arabic in the tenth century,2 and Abul Wefa (born 940 A.D.) wrote a commentary on it. Other commentaries appeared in the 11th century. All that Alkarkhi (c. 1010), for example, had in common with the method of the Hindus was in the solution of the equations Ay2 + 1 = x2 and a2y2 + C =x2 by the formulas 2m C- m2 Y = 2 - A' Y 2ma which conform, however, to the principles that he had found employed in many of the problems of Diophantus. It therefore seems doubtful that Alkarkhi knew the methods of the Hindus or that India was the source from which the Arab algebraists derived the theory of the solution of ay2 + by + c = x2.3 1 E. Strachey, p. 48. 2 H. Suter, "Die Mathematiker und Astronomen der Araber und ihre Werke," Abhandlungen zur Geschichte der mathematischen Wissenschaften, vol. X, p. 1-276, Leipzig, 1900. Mention is here made of over fifteen hundred Arabic scholars living between the years 750 and 1600. 3 F. Woepcke, "Extrait du Fakhri, Traite de'algebre par Abou Bekr Mohammed ben Alhacan Alkarkhi," p. 42, Paris, 1851. A Hochheim, "Kafi fil HIsab des Abu Bekr Muhammed Ben Alhusein Alkarkht, vol. II, p. 14, Halle. This work has no date imprinted, but it is probably 1877. 40 THE PELL EQUATION It is true that the approximation formulas for square root used by Alkarkhi and others starting with /a2 +r a+ 2a+ do not show any connection with the solutions of the simplest Pell equations but these were not the only approximations used by the Arabs. El-Hassar,1 in his extraction of an irrational root, says: "When it is asked what is the square root of 5, take the nearest square number in 5, this equals 4, subtract it from 5, the remainder is 1, divide this by 4, this gives 1/4, and add this to the root of 4, which equals 2, this gives 2}, and this is the approximate root of 5." This is the usual approximation a/a2 + r c a + r/2a, and 2a + r, 2a, is a solution of the Pell equation x2 - Ay2 = r2. In particular, the approximation obtained here, 9/4, gives x = 9, y = 4, as a solution of 2 - 5y2 = 1. El-Hassar then obtains a still closer approximation, 217, virtually by the use of the formula, -— r \2 +a2+r c a+r - 2 2 a+)2a thus giving another solution of the equation 2 - 5y2 = 1, namely x = 161, y = 72. The solution of X2 - Ay2 = r4, 1H. Suter, "Das Rechenbuch des Abu Zakarija el-Hassar," Bibliotheca mathematica, vol. II (3), p. 37, Leipzig, 1901. THE PELL EQUATION 41 by this method would be 8a4 + 8a2r + r2, 8a3 + 4ar. Alkalgadi1 used not only the above formulas, but also, when r > a, the formula2 a2 + r r a +2a r' As already stated, from the time of Heron, and even of Archimedes, the custom was quite general to attack the irrational square root through the relations a2 b ~ a = 2a ' a custom that obtained until the decimal fraction became well understood. When a more or less satisfactory first approximation was obtained the calculator proceeded methodically to obtain a better approximation by the same considerations that we use at the present time in the solution of the Pell equation. It will not be necessary to recount the different processes used by the great mathematicians in medieval times in carrying out this general method. Maximus Planudes,3 who flourished about 1325, gives general rules for finding approximate square roots; he gives, for example, elaborate computation for the square root of 235 obtaining the result 15k. This amounts to stating that x = 46, y = 3, is a solution of the equation X2 - 235y2 = 1. 1F. Woepcke, "Traduction du traite d'arithmetique d'Aboul Hasan Alt ben Mohammed Alkalcadi," Atti dell' Accademia pontificia de' Nuovi Lincei, vol. XII, p. 41, Rome, 1859. 2 F. Woepcke, "Recherches sur l'histoire des sciences mathematiques chez les orientaux," p. 36, Paris, 1855. 3 Maximus Planudes, "Das Rechenbuch des Maximus Planudes," translated from the Greek by H. Waschke, p. 48, Halle, 1878. 42 THE PELL EQUATION The method of Nicolas Chuquet,1 which was set forth in his "Triparty" in 1484, deserves attention. From two approximations to an incommensurable number, the one greater and the other less than the given incommensurable, he obtained a closer approximation. His method is one which leads to successive solutions of the Pell equation. No doubt Chuquet was incapable of demonstrating the validity of his method, and the Pell equation may never have been proposed to him, but in the examples which he gives he always stops at a remainder that has unity for the numerator; and as he calculates all these remainders it would have been easy for him to recognize experimentally the periodicity of these remainders. If po/qo, pl/ql, are two such approximations, then P' Po + Pi q' qo + ql is a new approximation; and it can be verified whether this is greater or less than the given incommensurable number. The new approximation can be used together with that one of the former approximations which is on the opposite side of the given incommensurable to get a yet closer approximation. In particular, approximations can be obtained in this manner to W/A, and these are evidently solutions of the equation X2 Ay2 1. This will be considered further in speaking of the work of 1 Nicolas Chuquet, "Le triparty en la science des nombres," Bullettino di bibliografia e di storia delle scienze matematiche e fisiche, vol. XIII, Rome, 1880. S. Giinther, "Analyse de l'ouvrage de Treutlein, Die deutsche Rechenkunst im XVI Jahrhundert," Zeitschrift fir das Realschulwesen, p. 430, 1877. M. Rodet, " Sur les methodes d'approximation chez les Arabes," Bulletin de la Societe Mathematique de France, vol. I, p. 162, Paris, 1879. P. Tannery, "L'extraction des racines carrees d'apres Nicolas Chuquet," Bibliotheca mathematica, vol. I (2), p, 17, Leipzig, 1887. THE PELL EQUATION 43 de la Roche,1 who copied from the manuscript of Chuquet in this matter as well as in many others. De la Roche2 in his "Larismetique" (1520), in setting forth a method which he calls the "Rigle de mediacion entre le plus et le moins," makes use of two series of fractions, 12345678 ~. never equal to 1; 2' 3' 4' 5 6' 7 8' neverequalto 1; l 1 '^T^ ~.. never equal to 0. 2' 3' 4' 5' 6' 7' 8' In finding the square root of 6, he begins with the assertion that -/6 is between 2 and 3. He then takes the first mean, 2~, and multiplies it into itself obtaining 6-, thus finding that the approximation is too large. He then selects from the table the next smaller fraction, obtaining 2 which multiplied by itself shows that the approximation is too small. He now finds a new approximation by forming a mean between 2~ and 2~ by adding numerators and denominators the result being 22. He tests the accuracy of this approximation for defect or excess and then gets a new one by adding numerators and denominators of 2~ and 2- as before, the result being 2f. In this manner the following approximations are obtained: 1 1 2 3 4 5 9 13 21 2- 2- 23 24 2, 2- 21 2' 3' 5' 7' 9' 11' 20' 29' 22 31 40 49 89 2- 2- 2 2 2 49' 69' 89' 109' 198 The more important of these for our purpose are 0 12 2+1 2+2, 1 A. Marre, "Notice sur Nicolas Chuquet et son triparty en la science des nombres," Bullettino di bibliografia e di storia delle scienze matematiche e fisiche, vol. XIII, p. 572, Rome, 1880. 2 M. L. Rodet, "Sur les methodes d'approximation chez les anciens," Bulletin de la Societe Mathematique de France, vol. VII, p. 159, Paris, 1879. 44 THE PELL EQUATION 4 0,0+4.1 9 1+2.4 9 + + 4.2' 20 =2 2+2.9' 40 4+4 9 89 9+2.40 2 89 9 +4.20' 2 198 2 + 2.89' By comparing these results with those obtained from the convergents when 16 is reduced to a continued fraction, 1 i6 = 2+ 1 2+4+, viz., 2 5 22 2 +4.5 49 5+2-22 1' 2' 9 1 +4-2' 20 2 + 2.9 218 22 + 4.49 89 9+4.20 ' ' we see that, by adding term to term in certain fractions, de la Roche's method amounts to an easy process of computing the convergents of /A and of calculating all the roots1 of the equation 2 - Ay2 = 1. Juan de Ortega2 also gives more or less evident indications of his use of the method of Chuquet. The rule for square root given by Ortega in the 1515 edition of his "Arithmetica" can be expressed by the formula /a2 + b bc a + 'b 2a +V' l If all the approximations in this particular example are taken they furnish solutions of the equation x2- 6y2 = B, where B repeats in order the six values, 1, - 5, - 6, - 5, - 2, 3. 2 Johane de Ortega, "Suma de arithmetica: geometria practica," p. 98, Rome, 1515; and "Sequitur la quarta opera de arithmetica et geometria facta et ordinata," p. 67, Messina, 1522. THE PELL EQUATION 45 which is the same as that of Alkarkhi, but in the later editions (1534, 1537, 1542), while there are some examples involving the formula of Alkarkhi, or rather of Heron of Alexandria, there are others which show that he possessed another method which plays an important role in the theory of numbers, for the following approximations,' all of which he gives, yield solutions to the Pell equation x2 - Ay2 = 1: 16 17 659 i128 c 11, 8iS0. 8 129c7 o 17 51' 18' 2820' 25 285 13 1300 3175 7 1 1375 19 135 11 103 109 68186 %175 c 8 156' 756 c 27 0, 611 c 24 8 151 197 1 a231 3 15 761, 800 o 28 169, 4100 c 64. 760 693' 32' Even if he had done no other work, these approximations would give the Spanish mathematician a worthy place in the history of mathematics. Buteo2 in his approximations to the square root of a number makes repeated use of methods which could be represented by the formula3 a2 +b \ a+ 2 For 166 he makes several approximations all of which 1 J. Perrot, "Sur une arithmetique espagnole du seizieme siecle," Bullettino di bibliografia e di storia delle scienze matematiche e fisiche, vol. XV, p. 163, Rome, 1882. 2 "Ioan. Buteonis Logistica, quae et arithmetica vulgo dicitur in libros quinque digesta," p. 76, Lyons, 1559. 3This method we have seen dates back to Archimedes and Heron We find it again in both letters of Nicolas Rhabdas. P. Tannery, "Notice sur les deux lettres arithmetiques de Nicolas Rhabdas (text grec et traduction)," p. 40, 68, Paris, 1886. Later examples of its use are seen in the works of Luca Paciuolo, Cataldi, Cardan, and Tartaglia. 46 THE PELL EQUATION give solutions to the equation x2 - 66y2 = 1, the last one being 81-2-, from which 8,4492 - 66.1,0402 = 1. Buteo also makes use of the method of Chuquet with the two series of fractions, the one ascending, the other descending, which have just been described. For /13 he gives the approximation 5, and this corresponds to the minimum solution of the equation x2 - 13y2 = - 1. Fermat1 was the first to assert that the equation x2 - Ay2 = 1 where A is any non-square integer, always has an unlimited number of solutions in integers. This equation may have been suggested to him by the study of some of the double equations of Diophantus;2 for Fermat says in a note on Diophantus, IV, 39, "Suppose, if you will, that the double equation to be solved is 2m + 5 = square, 6m + 3 = square. The first square must be made equal to 16 and the second to 36; and others will be found ad infinitum satisfying the equation. Nor is it difficult to propound a general rule for the solution of this kind of equation." By elimination the two equations just mentioned lead to 2 - 3y2 = - 12. "Oeuvres de Fermat publiees par les soins de MM. Paul Tannery et Charles Henry," vol. II, p. 334, Paris, 1894. 2 For a different view of Fermat's methods of solving the double equations see Paul von Schaewen, "Jacobi de Billy, Doctrinae analyticae inventum novum, Fermats Briefen an Billy entnommen," p. 61, Berlin, 1910. THE PELL EQUATION 47 In another place Fermat says he has discovered general rules for solving the simple and double equations of Diophantus. "Suppose for example, we have to make 2x2 + 7,967 = square. I have a general rule for solving this equation, if it is possible, or discovering its impossibility, and similarly in all cases for all the values of the coefficient of x and of the absolute term. Suppose we have to solve the double equation 2x + 3 = square, 2x + 5 = square. Bachet boasts in his commentary on Diophantus, of solving this in two particular cases; I make it general for all kinds of cases and can determine by rule whether it is possible or not." Fermat1 first proposed the general problem in a letter written to Frenicle in February, 1657, and in the same month he proposed it as a challenge problem, the third in his contest with the English mathematicians, Dr. John Wallis, Lord Brouncker, and others. The first two challenge problems were proposed at the same time, so that this one might be called the second challenge. Fermat's whole letter is worth quoting: "There is scarcely any one who states purely arithmetical questions, scarcely any who understands them. Is this not because arithmetic has been treated up to this time geometrically rather than arithmetically? This certainly is indicated by many works ancient and modern. Diophantus himself also indicates this. But he has freed himself from geometry a little more than others have, in that he limits his analysis to rational numbers only; nevertheless the Zetcica of Vieta, in which the methods of Diophantus are extended to continuous magnitude and therefore to geometry, witness the insufficient separation of arithmetic from geometry. 1 "Oeuvres de Fermat," vol. II, p. 333. 48 THE PELL EQUATION "Now arithmetic has a special domain of its own, the theory of numbers. This was touched upon but only to a slight degree by Euclid in his Elements, and by those who followed him it has not been sufficiently extended, unless perchance it lies hid in those books of Diophantus which the ravages of time have destroyed. Arithmeticians ['AptOLTr)Lciwv rakSes] have now to develop or restore it. "To these, that I may lead the way, I propose this theorem to be proved or problem to be solved. If they succeed in discovering the proof or solution, they will acknowledge that questions of this kind are not inferior to the more celebrated ones from geometry either for depth or difficulty or method of proof: "Given any number which is not a square, there also exists an infinite number of squares such that when multiplied into the given number and unity is added to the product, the result is a square." The problem thus set forth by Fermat is one of the most important steps in the history of the Pell equation. A freer translation of the Latin would read: For every given number which is not a square there exists infinitely many square numbers such that the product of each by the given number, with the addition of 1, is a square. Fermat illustrates his problem by a number of examples, one of which is as follows: "Given 3, a non-square number; this number multiplied into the square number 1 and 1 being added produces 4, which is a square. Moreover, the same 3 multiplied into the square 16 with 1 added makes 49, which is a square. And instead of 1 and 16, an infinite number of squares may be found showing the same property; I demand, however, a general rule, any number being given which is not a square. It is sought, for example, to find a square which when multiplied into 149, 109, 433, etc., becomes a square when unity is added." The above is very much like the problems set forth in THE PELL EQUATION 49 his letter to Frenicle in the same month (February, 1657), except in that place 109 is the only large number mentioned. These particular numbers, 109, 149, 433, are evidently chosen for their difficulty, for the fundamental solutions in these cases are very large numbers. In the conclusion of his letter to Frenicle, Fermat expressly says: " It appears without saying that my proposition looks only to finding integers which satisfy the question, for in the case of fractions the lowest type of arithmetician could see the solution."1 The secretary of Digby who made the copy for Lord Brouncker may have regarded the introduction, which expressly set forth the demand for integers, as unimportant, and so left it uncopied. This would explain why Brouncker, and after him Wallis, thought that only rational numbers were demanded.2 They accordingly first handled the problem by the method used in many places by Diophantus, and which also resembles one of the rules of Bhaskara. If the equation ay2 + 1 = X2 is to be solved for rational values, then the supposition that ay2 + 1 = (1 -ry)2 gives the equation 2r y r2 - a whence 4r2 = (r2 _ a)2 and if we substitute r2 = q, r2-a = d, we have 4q Y- = d2, 1 Oeuvres de Fermat," vol. II, p. 334. 2 G. Wertheim, "Pierre Fermats Streit mit John Wallis, ein Beitrag zur Geschichte der Zahlentheorie," Abhandlungen zur Geschichte der Mathematik, vol. IX, p. 563, Leipzig, 1899. 5 50 THE PELL EQUATION and this is the solution which Lord Brouncker sent to Paris. It was not until September 21, 1657,1 that Wallis received the problem through Brouncker and also the latter's solution. This solution had been transmitted indirectly to Fermat and had been so translated into French that Fermat failed to understand it. A few days after its receipt Wallis gave as his solution of the equation ay2 + 1 = x2 4:ps s =2 _ 4ap2' which was found by the method of Diophantus, and which, for p = 1, s = 2r, reduces to Lord Brouncker's expression. In his letter to Digby,2 dated June 6, 1657, Fermat says: "As to the question proposed in Latin, which I sent you, it also is in integers, therefore the solutions in fractions, which can be given at once from the merest elements of arithmetic, do not satisfy me." With this further knowledge of all the conditions of the problem Brouncker set to work again, and was soon able to communicate to Wallis3 the law of formation of an infinite product which for every value of the variable gives an integral solution of the equation Ay2 + 1 = x2 He gives this product for fifteen cases. For example, the product for 8y2 + 1 = x2 1"Commercium epistolicum," VIII, bound with J. Wallis's "De Algebra tractatus," p. 766, Oxford, 1685. "Oeuvres de Fermat," vol. III, p. 416, Paris, 1896. 2 "Oeuvres de Fermat," vol. II, p. 342, Paris, 1894. 3 "Commercium epistolicum," XIV, p. 775. "Oeuvres de Fermat," vol. III, p. 423, Paris, 1896. THE PELL EQUATION 51 is 1 5 29 169 1 x x 5 x 35 x 204 that is the values of y which make 8y2 + 1 a perfect square are 1 1 5 1 5 29 1, 1 X 55, 1 X5 X 5 I X 5iX 56 X 535, 1 5 29 169 1 X 5 X 5 X 5 5 X 524, * This product is evidently found through induction. The first two factors are found by experiment as integers satisfying the given equation. The remaining parts may be obtained according to the following rule: The numerator of each fraction is equal to the denominator diminished by the denominator of the preceding fraction, and each denominator is equal to the numerator of the improper fraction formed from the preceding mixed number. Lord Brouncker1 subsequently found that the smallest value of y was sufficient for determining all the others. If ay12 + 1 = x12 then a(2x1y1)2 + 1 is also a square. Indeed, a(2xlyl)2 + 1 = 4ayl2(ayl2 - 1) + 1 = (2ay12 - 1)2. From this he deduced other values of y, as follows: If Y = Yi, x = xi, is the fundamental solution of ay2 + 1 = x2, and if z1 is set for 2x1 then the values of y are Yl, ylZ, y l(z2 - 1), yl(zil - 2Z1), yl(Z14 - 312 + 1), y1(z15 - 4z13 + 3z1), y1(z16 - 5z14 + 6z12 - 1),.., "Commercium epistolicum," XVII, p. 789. "Oeuvres de Fermat," vol. III, p. 474. 52 THE PELL EQUATION in which the coefficient of the first term in parentheses is 1, of the second the series of natural numbers, of the third the triangular numbers, of the fourth the pyramidal numbers (the sums of the first n triangular numbers), etc. For computing these in the simplest manner Brouncker gave the following method, in which yi, y2, y3,... are the values of y: Y1. = y1, Y2 = ZlyX, Y3 = z1i2 - Y1, Y4 = z 3 - Y2, Y5 = Z1y4 - y3, '' In order to make a complete reply to Fermat's challenge, there was only needed a sure method for obtaining the fundamental solution, and this Brouncker succeeded in finding, as is set forth in the letters of Wallis under the dates December 17, 1657, and January 30, 1658.1 Euler2 has also given the same solution in his algebra. Stated in modern symbolism Brouncker's solution is as follows.3 Suppose a solution of the equation 2 - Dy2 = 1 exists, and let this solution be T, U. Then (1) T2 - DU2 = 1. It is required to find T and U. If p is the integer next smaller than /D, then it follows from the hypothesis that pU < T < (p + 1)U, and that T = pU + vi, 1 "Commercium epistolicum," XVII, p. 797, and XIX, p. 804. "Oeuvres de Fermat," vol. III, p. 490, 503. The most difficult problem solved here (p. 498), was x2 - 433y2 = 1, in which y has 19 figures. 2L. Euler, "Algebra," translated by J. Hewlett, 5th ed., p. 351, London, 1840. "Vollstandige Anleitung zur Algrabra," Part II, p. 226, Lund, 1771. 3 H. J. S. Smith, "Report on the theory of numbers," British Association report, p. 292, London, 1861; "Collected mathematical papers," vol. I, p. 193, Oxford, 1894. H. Konen, "Geschichte der Gleichung t2 - Du2 = 1," p. 39, Leipzig, 1901. THE PELL EQUATION 53 where v1 is an integer less than U. Substituting this value of T in (1), we have (2) Q(U, v) = 1, likewise in integers, and where Q signifies a homogeneous quadratic form. From (2) we have p1Il < U < (P1 + 1)vi, U = pliV + V2; V2 < Vi; Qi(vl, v2) = 1, and so on continually. Since by hypothesis, there is an integral solution of x2 - Dy2 = 1, and since the numbers, T, U V, vl, v2, * form a descending series T > U > vI > v2 **-, then sometime we must reach the equation Qn(vn, Vn+l) = 1, which gives Vn = PnVn+l; and, by reason of the hypothesis of the solvability of X2 - Dy2 = 1, this also gives an integral value of v,,+,. Through substitution we obtain the values of T and U in a manner similar to the algorism for obtaining the greatest common divisor of two integers. For example, consider the equation T2 - 13U2 = 1. We see that P = 3, whence 3U< T<4U; T= 3U+vl; Q(U, vl) = - 4U2 + 6Uvl + v12 = 1. 54 THE PELL EQUATION Hence, for P1 = 1, we have v1 < U < 2vl; U = v1 + V2; Ql(Vl, v2) = 312 - 2vv2 - 4v22 = 1. For P2 = 1, we have V2 < v1 < 2,2; V1 = V2 + v3; Q2(v2, v3) = 3v22 + 4v2V3 + 3v32 = 1. For P3= 1, we have v3 < v2 < 2v3; v2 = v3 -+ 4; Q3(v3, v4) = 4v32 - 2v34 - 3v42 = 1. For P4 = 1, we have V4 < V3 < 2V4; V3 = V4 + v5; Q4(v4, v5) = - v42 - 6v4v5 + 4v52 = 1. For P5 = 6, we have 6v5 < v4 < 7v5; v4 = 6v5 + V6; Q5(V5, V6) = 4v52 - 6v56 - v62 = 1. For P6= 1, we have v6 < v5 < 2v6; v5 = v6 + v7; Q6(v6, V7) = - 3v62 + 2v6v7 + 4v72 = 1. For P7 = 1, we have v7 < v6 < 2v7; v6 = V7 + V8; Q7(V7, V8) = 3v72 - 47V8 - 3V2 = 1. For P8 = 1, we have V8 < V7 < 2v8; v7 = Vs + v9; Q8(v8, Vg) = - 482 + 2V8v9 + 3v92 = 1. THE PELL EQUATION 55 For P9= 1, we have v9 < Vs < 2v9; v8 = v9 + vlo; Q9(v9, V10) = 92 - 69v10 - 41o2 = 1: from which v9 = 1, 10o 0. Therefore 8 = 1, V7 = 2, fV =3, v5= 5, 4 = 33, v3 = 38, v2 = 71, v = 109, U= 180, T= 649. Wallis adds, "In the same way it must be proceeded whatever number you wish that is not a square, being given."' It is interesting to note how completely this process conforms to that of continued fractions as later developed by Lagrange. For T VI vi 1 Vl V2 + V3 =3+ =3+ 1 1 =3+1+ 1 1+ 1+ 1+ v2 V3+V4 1+ 4 V3 and 13= (3; 1,1, 1,1, 6,...). Also, if we had not made v10 = 0 but had gone on as before, from P1o = 6, 1 "Commercium epistolicum," XIX, p. 804. "Oeuvres de Fermat," vol. III, p. 480. 56 THE PELL EQUATION we should have 6v0o < v9 < 7v0o; v9 = 6v0o + vll; Qlo(Vlo, vl) = - 4v1o02 + 6o10v11 + V112 = 1. The form Q0l is exactly like Q, and the whole round would be repeated in such a way that at Q20 a new solution would be found, and so on at Q30, Q40, ' *, a law that Lord Brouncker perceived, as we have just seen. Thus Fermat's challenge problem was solved, although the solution was not at once communicated to him. On December 1, 1657,1 Wallis wrote to Brouncker that Fermat had demanded not all the solutions, but only a large number of them, and that he would reserve the series which Brouncker had discovered and only communicate it to him when it was expressly demanded. At the same time2 he wrote to Digby that the restriction that a be not a square was not necessary, but that the given solution (in rational numbers) would avail in the special case when a was a square. He also wrote that the problem could be generalized to the equation ay2 + b2 = x by multiplying the solutions of the equation ay2 + 1 = X2 by b. As between Brouncker and Wallis, the credit for the solution seems to belong entirely to the former, as Wallis declares3 in a letter to Digby. Wallis also congratulates4 Brouncker that he has "preserved untarnished the fame which Englishmen have won in former times with French1 "Commercium epistolicum," XV, p. 776. "Oeuvres de Fermat," vol. III, p. 425. The date here is erroneously given as Oct. 1. 2 "Commercium epistolicum," XVI, p. 780. "Oeuvres de Fermat," vol. III, p. 430. 3 "Commercium epistolicum," XVIII, p. 798. "Oeuvres de Fermat," vol. III, p. 489. 4 "Commercium epistolicum," XIX, p. 802. "Oeuvres de Fermat," vol. III, p. 502. THE PELL EQUATION 57. men and has shown that England's champions in wisdom are just as strong as those in war." It was months before Fermat was informed of the details of the solution. This was partly on account of the first misunderstanding of the demand for integers, and partly because Wallis held back the complete solution after it was in his possession. Then, too, they always communicated through third parties. Fermat did not understand English, nor Wallis French. In the meanwhile Fermat wrote several letters to Frenicle renewing his demand for integers, and in Frenicle's letter1 to Digby, which reached Wallis February 20, 1658, it was stated that Fermat had solved the equation x2 - Ay2 = 1 for all non-square values of A up to 150, and that perhaps Wallis would be good enough to extend it to A = 200 or at least solve it for 151, although probably the case of A = 313 would be beyond his ability. To this last demand Brouncker replies in a letter2 to Digby which is dated March 13, 165k, in which he seems to wish it understood that the solution required but little time. Using his language and symbolism: "With the space of an hour or two at most this morning, according to the method therein delivered, I found that 313 X Q7,170,685 - 1 = Q126,862,368, and therefore that 313 X Q(2 X 7,170,685 X 126,862,368 =) 1,819,380,158,564,160 + 1 = Q32,188,120,829,134,849, which I thought fit to present to you that M. Frenicle may thence perceive that nothing is wanting to the perfect solution of that problem." 1 "Commercium epistolicum," XXVI, p. 821. "Oeuvres de Fermat," vol. III, p. 532. 2 "Commercium epistolicum," XXVII, p. 823. "Oeuvres de Fermat," vol. III, p. 536. 58 THE PELL EQUATION The solution of this challenge problem filled Frenicle and Fermat with great respect for Brouncker and Wallis.1 Fermat appears at one time to have been fully satisfied with the solution,2 but later he points out that although Brouncker and Wallis had given many particular solutions they had not supplied a general proof,3 meaning presumably a proof that the solution is always possible and that the method will always lead to the solution sought.4 Fermat nowhere publishes his solution. It seems very probable that it was much the same as Lord Brouncker's, and that if he had found a better solution he would have made it known. In fact Fermat is sometimes credited with Brouncker's solution, due to a misunderstanding of a sentence in Ozanam's Algebra, which in speaking of the Brouncker-Wallis solution, reads: "Une regle generale pour resoudre sette question, qui est de M. de Fermat." The ambiguity as to what the pronoun referred may have led Lagrange and others to suppose the rule came from Fermat.5 Without having seen the solution of Wallis and Brouncker, Malebranche, in 1658 or a little later, wrote an article upon the equation.6 He was always able to find a solution when the difference of the number given 1, Commercium epistolicum," XXXVIII, p. 846. "Oeuvres de Fermat," vol. III, p. 577. 2 "Oeuvres de Fermat," vol. II, p. 402. 3 "Oeuvres de Fermat," vol. II, p. 433. See also as to the contradictory opinions of Frenicle and an anonymous Latin writer, "Oeuvres de Fermat," vol. III, p. 592 and p. 605. In the last passage it reads, "Our analysts do not recognize a vestige of proof there." 4 T. L. Heath, "Diophantus," 2d ed., p. 287. 5 T. L. Heath, "Diophantus," 2d ed., p. 288. 6C. Henry, "Recherches sur les manuscrits de Pierre de Fermat suivres de fragments inedits de Bachet et de Malebranche," Bullettino di bibliografia e di storia delle scienze matematiche e fisiche, vol. XII, p. 696, Rome, 1879. This MS. is preserved in the Bibliotheque Nationale in Paris and is entitled "Essai de resolution par Malebranche de l'equation Ax2 + 1 = y2," Fonds Frangais, No. 25308, p. 9-56. THE PELL EQUATION 59 and a square is a factor of double the square root of the square. For example, if the given number is 33 or 39 of which the difference from 36 is 3, and this 3 is a factor of 12, which is double the square root of 36. For 39, say, 39x2 + 1 = 36x2 + 12x + 1. Therefore 3x = 12, x = 4, x2 = 16, and 39x2 + 1 = 625, etc. His manner of finding an infinity of solutions from a given one is perfectly general. The honor of having first recognized the deep importance of the Pell equation for the general solution of the indeterminate equation of the second degree belongs to Euler. He left several memoirs relating to this subject.' In a letter to Goldbach, August 10, 1732, Euler2 mentions the equation x2 - Ay2 = 1 as necessary in order to make ay2 + by + c a square, and goes on to say: "Problems of this kind have been discussed between Wallis and Fermat.... The most difficult example was to find numbers which put in place of y would make 109y2 + 1 a perfect square. Dr. Pell3 1L. Euler, "De solutione problematum Diophantaeorum per numeros integros," Commentarii Academiae scientiarum imperialis Petropolitanae, 1732, vol. VI, p. 175, St. Petersburg, 1738. "De resolutione formularum quadraticarum indeterminatarum per numeros integros," Novi commentarii Academiae scientiarum imperialis Petropolitanae, vol. IX, p. 3, St. Petersburg, 1764. "De usu novi algorithmi in problemate Pelliano solvendo," Novi commentarii Academiae scientiarum imperialis Petropolitanae, vol. XI, p. 28, St. Petersburg, 1765. 2 P. H. Fuss, "Correspondance mathematique et physique de quelques celebres geometres du XVIIIieme siecle," p. 37, St. Petersburg, 1843. 3 Note here probably the first reference of the equation to Pell. For a suggestion that this mistake of Euler was due to cursory reading, note also that Euler makes a mistake as to the most difficult cases solved by Wallis and Brouncker, these really being the equations 433y2 + 1 = x2 and 313y2 + 1 = x2 60 THE PELL EQUATION an Englishman devised a unique method of solving problems of this kind as shown in the works of Wallis... according to which at the beginning of my work I have that 1 + aX2 makes a square. But this method was only available for numerical numbers, and not for formulas having arbitrary coefficients." In the same year,1 Euler set forth a method for reducing the solution of ay2 + by + c = x2 to that of the Pell equation x2 - Ay2 = 1. He gave a list of solutions up to A = 68. He also proved that the successive solutions of ay2 + by + c = 2, when one is known, require that one solution of av2 + 1 = u2 must also be known. Given one value of y, say n, which makes ay2 + by + c a perfect square, and one value of v, say q, which makes av2 + 1 a perfect square, that is, when (1) an2 + bn + c = m2 and aq2 + 1 = p2, Euler finds any number of solutions of ay2 + by + c = x2 and the law for forming them. He then takes the particular case ay2 + by + d2 = x2 where (since y = 0, x= d L. Euler, "De solutione...," loc. cit. A. Aubry, "L'oeuvre arithm6tique d'Euler," L'enseignement mathematique, vol. XI, p. 329, Paris, 1909. T. L. Heath, "Diophantus," 2d ed., p. 288, Cambridge, 1910. THE PELL EQUATION 61 satisfies the equation) we can substitute 0 for n and d for m in (1). Then again putting b = 0 and d = 1, he is in a position to write down any number of successive solutions of av2 + 1 = u2 when one solution, v = q, u = p, is known. The values of v are O, q, 2pq, 4qp2 - q,.. and the corresponding values of u are 1, p, 2p2- 1, 4p3-3p, *, the law of formation being that if A, B, are two consecutive values in either series, then the next following value is 2pB - A. But how is the fundamental solution, p, q, of the Pell equation to be found? Euler first points out that when a has one of many particular forms, the values, p, q, which satisfy the equation, can at once be written down. The simplest of these forms are found when a is one more or one less than a perfect square. The following are the cases mentioned by Euler: a = e2- 1, q = 1, p = e; a = e2 1, q = 2e, p = 2e2 +1; a = a2e2b 2aeb-, q = e, p = aeb+l 1 (where a may even be fractional provided aeb-l is an integer); a = (aeb + 3eu)2 + 2aeb-1 + 23eu-1, q = e, p = aeb+l + eu+l + 1; a = -aCk2e2b aeb-l, q = ke, p = 1 ak2eb+l - 1. 62 THE PELL EQUATION Barlow1 calls attention to the fact that Euler might have added among the simplest of these relations, 2e a=e2 —, q=c, p =e 1. C If a can not be put in any of these forms Euler says that the method explained by Wallis must be used. Euler goes on to remark that the above procedure for finding successive solutions to the equation av2 + 1 = U2 gives a very easy way of finding closer and closer approximations to the value of the square root of a. It was not until 1765 that he showed the reverse process, that the convergents in the approximation of -a lead to the solutions of the equation u2 - av2 = 1. Since aq2 + 1 = p2 we have = p2 - l q and if p (and therefore q) is large, p/q is a close approximation to aa, the error being not greater than 1 2q 4a Euler illustrates by taking a6, the fundamental solution of 6v2 + 1 = U2 being p=5, q=2. Taking the values above given and substituting for p and q their values, the successive corresponding values P, Q, of u, v, respectively become, P = 1, 5, 49, 485, 4,801, 47,525, 470,449, 4,656,965, ~ *, Q = 0, 2, 20, 198, 1,960, 19,402, 192,060, 1,901,198, **, P. Barlow, "Euler's algebra," translated by Hewlett, note on p. 359, London, 1840. THE PELL EQUATION 63 and the successive values of P/Q are closer and closer approximations to 6. It will be noted that this method of finding approximations to!a is the same as that which, according to the hypothesis of Zeuthen and Tannery, Archimedes used for ~/3. In 1759 Euler1 stated the theorem that the product of two expressions of the form x2 + Ay2 is of the same form; but Goldbach seems to have first remarked this important theorem in' a letter to Euler in 1753. If we have the identity (a2 + Ab) (a2 + Ab2) = (aa =+- Abf3)2 + A(aO3 ba)2, it follows that if a, b, is one solution of the Pell equation x2 - Ay2 = 1 and a, 3, is a solution of the equation x2 - Ay2 = B, then we will have x = aa Ab/3, y = a3: ba, as another solution of the latter equation. But we have seen that the Hindus knew this remarkable formula at least a thousand years earlier, so that it was merely a case of independent rediscovery. Euler, in letters2 to Goldbach in 1753 and in 1755, mentions certain improvements on the Pellian methods; and in his next paper3 on this subject, he again speaks of finding all the solutions of ay2 + by + c = X2, or of ay2 + b = x2, when one solution is known. In the discussion of the 1L. Euler, in Novi commentarii Academiae scientiarum imperialis Petropolitanae, 1759, vol. VII, p. 3, St. Petersburg, 1761. 2 P. H. Fuss, op. cit., p. 614, p. 629. 3 L. Euler, "De resolutione...," loc. cit. 64 THE PELL EQUATION latter equation he deduces "the remarkable theorem which contains within it the foundations of higher solutions." This theorem is the same as that which the Hindus had used and resembles the one mentioned above. If a, b, is a solution of the equation x2 - Ay2 = SI and a, 3, is a solution of the equation x2 - Ay2 = S2, then x aa -b Abe, y = a: =~ ba, satisfies x2 - Ay2 = SS2. A more important paper1 of Euler's was published in 1767 in which he solved the equation x2 - Ay2 = I by reducing the method of Brouncker to the development of -A7 into a continued fraction. He showed this development to be symmetric and periodic. He takes up again the equation ay2 + by + c = x2, but goes on to state that "this investigation can be extended to any quadratic equation between two numbers, Ay2 + 2Bxy + Cx2 + 2Dy + 2Ex + F = 0, if one solution is known." Euler does not conceal the fact that the calculation is complicated and laborious. Solutions of the Pell equation are here given for values of A from A = 2 to A = 99 and also for A = 109, 113, 157, 367. He goes on to show that the labor of its solution can be significantly lightened by the development of IA into a continued fraction, a process probably the reverse of that of Archimedes. 1 L. Euler, "De usu novi algorithmi in problemate Pelliano solvendo," Novi commentarii Academiae scientiarum imperialis Petropolitanae, 1765, vol. XI, p. 28, St. Petersburg, 1767. THE PELL EQUATION 65 Euler begins by asserting without proof that if p, q, is a solution of the equation 2 - Ay2 = 1, then P>, q and p/q gives an approximation so close to the value of 4A that a more exact one cannot be made without bringing larger numbers into the development. He sets forth the development by continued fractions first for the particular examples, -/13, 461, -67, and then in a general form. If /z be the given surd and v the greatest integer in -lz, then z = v + - a+ 1 b + C + — c+d+ If -=v- +~, and v=A, Xv then z+A lz+A z - A2 o a where a = z - A2. If a is the greatest integer in ( lz + A) /a, we have v +A a< and we set in the second place x = a+y then since z = a + A2 6 66 THE PELL EQUATION we have,/z-A + aa iz + B Y 1 + 2aA - a2a = 3 where B = aa-A and = 1 -a(A-B). Now let b be the greatest integer in (- + B) /3, so that v+B b<, and we set in the third place y = b+ w' and proceeding in a similar manner we obtain the following table1 of Euler: Capiatur. Tum Vero. Eritque. _v~A I A = v, a = z-A2 =z-v2, a-, z-B2 v + B II B = aa-A, 3 = = 1 + a(A -B), b, a t z-C2 v+C III C = 3b - B, y a= +- b(B - C), c v -, z -D 2 v + D IV D = yc-C, 8 = = + c(C - D), d-,V D z- E2 v + E V E = 5d-D, e = Z y d(D -E), e + etc. etc. etc. The sign of equality is to be taken in the last column only when the fraction represents an integer. Since _ v+A A = v, and a < v, 1 Similar tables are found in many algebras since this time. For example, see I. Todhunter, "Algebra for the use of colleges and schools," p. 379, London, 1870. THE PELL EQUATION 67 we have B = a -A - v, > 1, b6 2v, and C = b - B v, y 1, c < 2v, etc., the integers A, B, C, D,... < v, the integers,and the indices,, 1, and the indices a, b, c, d,... = 2v. Euler declares that after the index 2v is reached, the values a, b, c, * * repeat themselves and the whole development begins anew, but he gives no proof that the index 2v exists. In examples he shows that a, b, c, *.. and a, f3, y,.. are repeated periodically, and for seven or eight different forms of z like those mentioned on page 61 general values for the indices and the Greek letters are given. The successive convergents to the continued fractions are then investigated. They are developed according to the law apparent in the following series, v a b c *. m n * 1 v av+ 1 (ab + l)v + b M N nN+M ' 1' a ' ab + 1 ' ' P' Q' nQ+P' There is a shorter algorism introduced for the convergents as follows: 1 (v) (v, a) (v, a, b) (v, a, b, c) (v, a, b, c, d) O' 1' (a) ' (a, b) ' (a, b, c) ' (a, b, c, d) where (v, a) = a(v) + 1; (v, a, b) = b(v, a) + (v); (v, a, b, c) = c(v, a, b) + (v, a); *. (a) = al1 + 0; (a, b) = b(a) + 1; (a, b, c) = c(a, b) + (a); *-*, Euler states that he had proved the following transforma 68 THE PELL EQUATION tions, in which the functions are called cumulants. They can be readily verified. (v, a, b, c, d, e) = v(a, b, c, d, e) + (b, c, d, e), v(a, b, c, d, e) = (v, a)(b, c, d, e) - v(c, d, e), (v, a, b, c, d, e) = (v, a, b)(c, d, e) + (v, a)(d, e), (v, a, b, c, d, e) = (v, a, b, c)(d, e) + (v, a, b)c. By the help of these formulas we can save ourselves the reckoning of practically half of the convergents. If, for example, when -iz is developed into a continued fraction the indices v, a, b, c, c, b, a, 2v, are obtained, 2v closing the period, and we take (a, b, c) = (c, b, a) and designate the convergents by xi/yi, x2/y2, ~* *, x8/y8 where xl/yl = 1/0 we find that x8 = xSys + x4y4, Ys = Y52 + Y42, where x5 = (v, a, b, c), 4 = (v, a, b), y5 = (a, b, c) = (c, b, a), y4 = (a, b) = (b, a). We do not need to reckon x6, X7, y6, and y7. Generalizing this relation we have Y2m = Ym+12 + Y/m2, X2m = Xm+lYm+l + XmYm. This occurs, however, in the case of the unusual period having the double middle term and an odd number of terms. When there is a single middle term and an even number of members in the period, we have Y2m-1 = Ym(Ym+l + Ym-1), X2m-1 = Xm+lYm + XmYm-1. When m and z are large numbers, these formulas allow an immense saving in the labor of the computations of the solutions. Konen1 quotes the proof of these formulas 1 H. Konen, "Geschichte der Gleichung t2 - Du2 = 1," p. 56, Leipzig, 1901. Konen quotes G. W. Tenner, "Einige Bemerkungen iuber die Gleichung t2 - Du2 = -~ 1," Programm, Merseburg, 1841. THE PELL EQUATION 69 and says that they have been worked out independently by G. W. Tenner, and many writers ascribe them to the latter. It is now shown that ifx = 1, y = 0, then.x2 = zy2 + 1; if x = (v), y = 1, then x2 = zy2 - a; if x = (v, a), y = (a), then x2 = zy2 -+; if x = (v, a, b), y = (a, b), then x2 = zy2 - 7; if x = (v, a, b, c), y = (a, b, c), then x2 = zy2 + 5; if x = (v, a, b, c, d), y = (a, b, c, d), thenx2 =zy2 - e; If one of the letters 3, 5,. = 1, we have a solution of the equation x2 - y2 = 1. But none of these letters can equal =- 1 unless at the same time the corresponding index is 2v. If then any period contains the index 2v and we place x, y, equal to the convergent values which correspond to this period, we obtain x2 - zy2 = 1, if the number of indices in the period is even; and x2 - zy2 = - 1, if the number is odd. In the first case we have directly the solution sought; in the other case by taking two periods' together, we get a period having an even number of closing mem1There is an error on this point in M. Cantor, "Vorlesungen fiber Geschichte der Mathematik," vol. IV, p. 159, 3d ed., Leipzig, 1908; for there it says two periods further and convergents at the end of the third period. By combining two periods with an odd number of indices a period is made having an even number of indices and closing with 2v, which would therefore furnish a solution of the Pell equation x2 - zy2 = 1. But if as the text directs, we proceed two periods further than the close of the first odd period we are at the close of another odd period and do not have a solution of the equation x2 - zy2 = 1 but of the equation x2 - zy2 = - 1. To get the solution of x2 - zy2 = 1 we must not take the convergent at the close of the third period, as the text says, but at the end of the second or fourth or some even number of periods. If we took the convergent at the end of the third period we would only have the solution of x2 - zy2 = - 1. 70 THE PELL EQUATION bers with 2v. Then the values of x and y corresponding to this second index 2v are taken. Or, if from the equation x2 - zy2 = - I we take xi = 2x2 + 1, yi = 2xy,we have a solution, xi, yl, of the equation xi2 - zy12 = 1. These procedures lead in a convenient manner to the solution of the equation x2 - zy2 = 1 in least integers. Care should be taken not to overlook the solution of x2 - zy2 = -1 in least integers in the particular case in which z = a2 + 1, in which case the equation is satisfied by x = a, y = 1. As already stated, Euler does not prove that we will ever arrive at the index 2v, and unless this is shown we are not sure that we will reach any solution beyond x = 1, y = 0. It was left to Lagrange to clear this matter up. H. J. S. Smith,1 one of the greatest English authorities on the theory of numbers, says: "Euler observed that x/y is necessarily a convergent to the value of A/z, so that to obtain the numbers x and y it suffices to develop i/z in a continued fraction. It is suggestive, however, that it never seems to have occurred to him that to complete the theory of the problem, it was necessary to demonstrate that the equation is always resoluble, and that all its solutions are given by the development of the a/z. His memoir2 contains all the elements necessary to the demonstration, but here, as in some other instances, Euler is satisfied with an induction which does not amount to a rigorous proof." The following opinions may help us to estimate properly the work of Fermat, Brouncker, and Euler, and they form an appropriate introduction to the important additions of Lagrange. Legendre3 says: "Fermat is the first who seems to 1 H. J. S. Smith, "Report on the theory of numbers," British Association report, p. 135, London, 1861; "Collected works," vol. I, p. 194, Oxford, 1894. 2 L. Euler, "Commentari arithmetical" vol. I, p. 316, St. Petersburg, 1849. 3 A. M. Legendre, "Theorie des nombres," 3d ed., ~ 36, Paris, 1830. THE PELL EQUATION 71 have known the solution of the equation x2 - Ay2 = 1. At least he proposed this problem as a challenge to the English mathematicians. It is Lord Brouncker's solution which we find in the works of Wallis and which is contained almost word for word in the second part of Euler's algebra. But on the one hand Fermat has made public nothing of his own solution, and on the other hand the process of the English mathematicians, although it is very ingenious, does not show in a definite manner that the problem is always solvable. It remained therefore to prove that the equation x2 - Ay2 = 1 can always be solved in integers. Lagrange1 has done this in a sagacious as well as rigid manner.... This proof, as well as the one added by him, must be considered the most important step which has been made up to the present time in the indeterminate analysis." Gauss2 expressed himself in the following manner: "This famous problem (to solve in integers all indeterminate equations of the second degree) Lagrange3 has completely solved. There is also an inferior complete solution by Lagrange in the supplement to Euler's algebra. The treatise of Lagrange grasps the problem in its entire generality and in this connection leaves nothing to be desired." Lagrange4 himself sums up his opinion in the following words: "It is in truth, very surprising that M. de Fermat who has been for so long a time and with such success occupied with the theory of integral numbers has not sought to solve generally the indeterminate problem of the second and higher degrees, as M. Bachet 1 J. L. Lagrange, "Solution d'un probleme d'arithmetique," Miscellanea Taurinensis, vol. IV, p. 41, Turin, 1766. "Oeuvres de Lagrange," vol. I, p. 671, Paris, 1867. 2 C. F. Gauss, "Disquisitiones arithmeticae," ~ 222, p. 309, Leipzig, 1801. 3 J. L. Lagrange, "Histoire de l'Academie de Berlin," p. 165, 1767, p. 181, 1768. 4 J. L. Lagrange, Memoires de l'Academie de Berlin, vol. XXIV, p. 236, 1768. 72 THE PELL EQUATION has done with that of the first degree. We must believe that he has applied himself to this research, for the problem which he proposed to M. Wallis and all the English geometers, is the keystone of the general solution of these equations. The problem is a particular case of equations of the second degree in two unknowns, and consists in finding two integral squares of which the one being multiplied by any given non-square integer and then subtracted from the other the remainder should be equal to unity. Whether M. Fermat has not continued his researches in this matter, or a record of them has not come to us, it is certain that no traces of them have been found in his works. It appears also that the English geometers who have solved the problem of M. de Fermat have not known all its importance for the general solution of indeterminate problems of the second degree. At least, we do not see that they have ever made use of it and Euler is, if I do not mistake, the first who has shown how by aid of this problem we are able to find an infinity of solutions in whole numbers of all the equations of the second degree in two unknowns of which we already know one solution. Since the work of M. Bachet (1613) up to the present time, with the exception of the memoir which I gave the past year upon the solution of indeterminate problems of the second degree, the theory of this sort of problems has properly speaking not been pushed beyond the first degree." The first admissible proof of the solvability of the equation x2 - Ay2 = 1, as has been stated, was given by Lagrange.1 He shows that in the development of 4A we shall obtain an infinite number of solutions of certain equations of the form x2 - Ay2 = B, and that by multiplying a sufficient number of these equations together, member by member, we can deduce the solution of x2 - Ay2 =1. Lagrange was dissatisfied with his firstJ. L. Lagrange, Miscellanea Taurinensia, loc. cit. THE PELL EQUATION' 73 proof and soon gave a simpler and more direct method which depended on the completion of the theory left unfinished by Euler of the development of a quadratic surd into a continued fraction. This new proof showed directly that x2 - Ay2 = 1 is solvable for every nonsquare integer A, which was shown in a very circuitous manner in the first proof. The new proof also served for the solution of x2-Ay2 =B for all values of IB < /A when such solution exists. This treatise1 by Lagrange gave new and general methods for the solution of equations of the form x2- Ay2 B, and in general of every equation of the second degree in two unknowns, not only in integers but also in mixed numbers. Lagrange showed how to reduce the general equation to the form x2 - Ay2 = B. He then discussed the cases in which this equation is solvable, first considering that in which x and y are either mixed numbers or integers, and afterwards that in which only integers are permissible. After considering the possibilities when IBI < I/A, he discussed the general case in which B is any integer.2 He also discussed the case in which A is a negative number. Throughout the work there are numerous examples which help to make clear the general proofs. Lagrange then comes to the general Pell equation, which he writes in the form - 1 = r2 - Bs, where B is a positive non-square integer. It is taken as a special case of - E = r2 - Bs2, 1 J. L. Lagrange, "Sur la solution des problemes indetermines du second degre," Histoire de l'Academie de Berlin, vol. XXIII, p. 272, vol. XXIV, p. 236, Berlin, 1767, 1768. "Uber die L6sung der unbestimmten Probleme zweiten Grades," translated in German and edited by E. Netto, Oswald's Classics, No. 146, p. 104, Leipzig, 1904. 2 For a modern discussion of this case, see Chrystal's Algebra, Part II, 2d ed., p. 482, London, 1906. 74 THE PELL EQUATION where I E < I 'B. The procedure in outline is as follows:l Letting E = 1, we seek first a positive integer E such that /B - 1 < e < /B, and for which B - e2 is divisible by 1. This goes without saying, of course, but this is only a special case of B - e2 divisible by E where E may be different from 1. Therefore in this case we must take for c the integer immediately below iB. As we now know E and E, we may obtain the series, E, E1, E2, -"; el, E2,,; Xi, X2, Xs, " *; by the aid of the following formulas,2 EE1 = B - e2, EIE2 = B - E2, E2E3 = B - E22, E3E4 = B - e32, (e1 = X1E1 - e, e2 = X2E2 - 1, e3 = X3E3 - 2, > X1 > > 1, B+e E1 IB+ E1 B_ - +l >,2 > B - _ 1, E2 E2 JB -+ E2 > X3 > B -+ 2_ 1, E3 E3 We continue in this construction until two members, E, and E,+l are found which are identical with E and E1. The proof that this always occurs is given at length.3 1 J. L. Lagrange, "Uber die L6sung..," loc. cit. 2 Op. cit., p. 48, p. 52. 3 Op. cit., p. 55. THE PELI EQUATION 75 Then he finds X and Y, called first by him the fundamental solution, such that X = EkA-,_ + 10-2, Y = kl, in which the k's are obtained from the following formulas,' k = 1, kl = Xlk, Z 2 = X2kl + k, k = X3k2 + kl, k4 = X4k3 -+ k2, k5 = X5k4 + k3, This solution, X, Y, satisfies the equation X2 - BY2 = i 1. In general, (X + Y rB)n + (X - Y B)n r= 2 2 B in which n must be so taken that nu is even or odd according as r2 - Bs2 = + 1or - 1. When r2 - Bs2 = 1, ngu must be even. If,u is even we may take any positive integer for n; if,u is odd we may choose only even numbers for n. Accordingly every equation of the form r2 - Bs2 = 1 is solvable in integers. For the equation r2 - Bs2 - 1 to be possible Lagrange stated the theorem that B could have no prime factors except those of the form 4n + 1. This condition is necessary, for otherwise B could not be a divisor of x2 + 1; but is not sufficient, since, for example, there is 1 Op. cit., p. 64. 76 THE PELL EQUATION no solution when B equals 5.41. If r2 - Bs2 = - 1, nu must be odd; and this is only possible if u is also odd. In case u is an even number this equation is never solvable in integers. If, however, u is odd the equation can be solved by the formulas just given, by substituting an odd number for n. In his Additions1 to Euler's Algebra Lagrange also gave complete solutions of the equations 2 - By2 = 1 and x2 - By2 = E. The two indeterminate equations x2 - Ay2 = 1 and x2- Ay2 = 4 are of primary importance in the theory of quadratic forms of a positive and non-square determinant. When the complete solution of these equations is known we can deduce from the single representation of a number by a form every representation of the same set, and from a single transformation of either of two equivalent forms into the other, every similar transformation. Gauss has transformed the problem of the Pell equation by his method of substitutions. He has avoided the use of continued fractions and has shown that if we form by the method he indicates the period of a quadratic form of determinant A we may infer the complete solution of the equations x2 - Ay2 = 1 and x2 - Ay2 = 4 from the automorphics of any reduced form, according as the form is properly or improperly primitive.2 The following theorem from the Disquisitiones Arithmeticae3 shows how he connected the Pell equation with the theory of quadratic forms. 1 J. L. Lagrange, "Euler's algebra with additions by M. de la Grange," translated by Hewlett, Chap. VIII, p. 578, London, 1840. On this page, through a misunderstanding of Ozanam, Lagrange seems to think that the solution of Lord Brouncker should be credited to Fermat. 2 C. F. Gauss, "Disquisitiones arithmeticae," ~~ 198-202, p. 259-273, Leipzig, 1801. 3 Op. cit., ~ 162. "Werke," vol. I, p. 129, 2d ed., G6ttingen, 1870. THE PELL EQUATION 77 If the form AX2 + 2BXY + CY2 = F implies the form ax2 + 2bxy + cy2 = f, and if a certain transformation is given which transforms the first into the second; from this to deduce all the transformations which produce the same result. The solution is substantially as follows: If the given transformation is X = ax + py and Y = yx + by, we first let another similar transformation produce the same result. Let this new transformation be X = a'x + b'y, Y = y'x + 6'y. Let the determinants of the forms F and f be D and d, and let ca -,y = e, a'8' - '^y' = e'. Then d = De2 = De'2, and since by hypothesis e and e' have the same sign, it follows that e = e'. We have moreover the following six equations: (1) Aa2 + 2Bay + Cy2 = a, (2) Aa'2 + 2Ba'y' + Cy'2 = a, (3) AaO + B(a5 + y7) + Cya = b, (4) Aa'/' + B(a'a' + 3'y') + Cy'a' = b, (5) A32 + 2B13 + C2 = c, (6) Af3'2 + 2Bf3'a' + C'2 = c. If for the sake of brevity we designate the numbers Aaa' + B(ay' + a'y) + Cyy', A(aO3' + Oa') + B(a3' + Oy' y' + 6a') + C(yb' + 5y'), A33' + B(/6' + 53') + C6S' by a', 2b', c', respectively, we deduce from the preceding equations the following: (7) a'2- -D(ac' - ya')2 = a2 (8) 2a'b' - D(ay' - ya')(a/ + ry' - ' - 5a') = 2ab, 4b'2 - D[(a/5' +,y' - 7/3' - 6a')2 - 2ee'] = 2b2 + 2ac. By adding 2Dee' = 2d = 2b2 - 2ac to the equation above we have (9) 4b'2 - D(aS' + y,' - 7y3' - ba')2 = 4b2, 78 THE PELL EQUATION whence a'c' - D(a6' - y') (' - 6a') = b2, and by subtracting D(a5 - 3y)(a'a' - 1'y') = b2 - ac from this equation, we have (10) a'c' - D(ay' - ya')(f3' - 53') = ac, and we also obtain (11) 2b'c' - D(a'+ f3y'- 7/' - &6')(3' - 6/3') = 2bc, (12) c'2 - D(03' - /3')2 = c2. Let now the greatest common divisor of a, 2b, c be m, and from this assumption determine numbers 2f, A3, (, so that 3Ia + 23b + ~ c = m.1 Multiply equations (7), (8), (9), (10), (11), (12) by 32, 2Nf8, $32, 22S, 23(, G2, respectively, and for the sake of brevity let (13) Xa' + 23b' + Cc' = T, and (14) 2(ac' - ya') + (aS' + y' - y' - 6') + G(~8' - a:') = U where T, U, manifestly are integers. It is easily shown that T2 - DU2 = m2. We are then led to the elegant conclusion that from any two similar transformations which transform F into f, there is found a solution in integers of the indeterminate equation t2 - Du2 = m2, namely, t = T, u = U. Gauss then goes on to show that from one transformation and one or more solutions of t2 - Du2 = m2, other solutions may be found. 1 C. F. Gauss, op. cit., ~ 40. THE PELL EQUATION 79 Dirichlet,1 after a brief review of Legendre's2 methods as applied to the Pell equation, develops a number of theorems on the solvability of x2 - Ay2 = - 1 for certain values of A. If p, q, is the solution of x2 - Ay2 = 1 in smallest integers (except 1, 0), then (p + 1)(p- 1) = Aq2. It should be observed that p + 1 and p - 1 are relatively prime numbers when p is even, and merely have the common factor 2 when p is odd. In the first case we have the following results: p + 1 = Mr2, p- 1 = Ns2, A = MN, q =rs; and in the second case we see that p + 1 = 2Mr2, p- 1 = 2Ns2, A = MN, q = 2rs, where M, N, and therefore r, s, are completely determined by p. Furthermore M and N are in the first case the greatest common divisor of A and p + 1, and of A and p- 1 respectively; and in the second case of A and (p + 1)/2, and of A and (p - 1)/2 respectively. From these equations we have, for the first and second cases respectively, Mr2 - Ns2= 2, Mr2 - Ns2 = 1. From this beginning Dirichlet worked out a large number of theorems of which the following are typical: For every prime number A of the form 4n + 1 the equation x2 - Ay2 = - 1 is always possible-a fact that had already been shown by Legendre. For every prime number A of the form 8n + 7 the equation x2 - Ay2 = 2 is always possible, and for every prime number A of the form 8n + 3 the equation x2 - Ay2 = - 2 is likewise 1 G. L. Dirichlet, "Einige neue Satze uiber unbestimmte Gleichungen," Abhandlungen der Koniglich-Preussischer Akademie der Wissenschaften, p. 649, Berlin, 1834; "Werke," vol. I, p. 221, Berlin, 1889; "Zahlentheorie," p. 202, 4th ed., Braunschweig, 1894. 2 A. M. Legendre, "Theorie des nombres," 2d ed., p. 54, Paris, 1808. 80 THE PELL EQUATION always possible. If a and b are primes of the form 4n + 1 the equation x2 - Ay2 = - 1 is possible when (1) A = a2k+ (2) A = 2a, a= 8n + 5, a-1 (3) A = 2a, a = 16n + 9, 2 4 -1 (moda), (4) A = ab, (b) + 1, ()= 1 - 1 ( =-1. The conditions (3), (4), (5), are not all necessary. There are also theorems like the following: If a, b, c, are prime numbers of the form 4n + 1, such that (b/a) = (c/a) = 1, (b/c) = - 1, and at the same time (bc/a)4 = - 1, and (alb)4(a/c)4 = - 1, then the equation x2 - aby2 =- 1 is solvable. For example, these relations are fulfilled when a = 5, b = 41, c = 109; for we have (415 (109 1 109) (41 1 5 i 4 1 109 5 -1 The equation x2 - 5.41-109y2 = - 1 is therefore solvable. Dirichlet also generalizes the relations p2 _ Aq21 2 = 12, Mr - Ns2 = 2 2 = 1, so as to make them applicable to any solution, P, Q, as well as merely to the fundamental one, proving that P - p1 (mod A), and for n odd pn - 1 (mod A) or (mod 2A), according as p is even or odd. (b/a) is the Legendre symbol which signifies that b is a quadratic residue or non-residue of a according as it is set equal to + 1 or - 1. If c is a prime number of the form 4n + 1 and if k is not divisible by c, and (k/c) = + 1, that is if k(c-l)2 = 1 (mod c), then either k(c-l)/4 = 1 or k(c-'i) - -1 (mod c). This remainder + 1 or -1 is designated by (k/c)4. THE PELL EQUATION 81 Along the line suggested by the work of Gauss, Dirichlet1 developed the following theorem which shows the intimate connection of the equation with the theory of quadratic forms: If (a, b, c) is a form2 of determinant D and divisor o-, and if (k) is any proper substitution through which (a, b, c) is transformed into itself, then always t - bu - cu au t + bu X = -, I =, v=-, p= = -, where t, u, are integers which satisfy the indeterminate equation t2 - Du2 = o-2; and conversely from every solution, t, u, of the equation, the above formulas afford a substitution through which the form (a, b, c) is transformed into itself. From the theory of quadratic forms,3 we have the following equations: (1) Xp - Jv =- 1, (2) a2 + 2bXv + cv2 = a, (3) aXgy + b(Xp + pv) + cvp = b, (4) from (1) Xp = Av + 1, 1 G. L. Dirichlet, "Vorlesungen ilber Zahlentheorie," 4th ed. (edited by Dedekind), p. 149, p. 201, Braunschweig, 1894. See also Dirichlet, "Recherches sur les forms quadratiques a coefficients et a indeterminees complexes," Werke, vol. I, p. 535-618, Berlin, 1889; Journal fiur die reine und angewandte Mathematik, vol. XXIV, p. 291, Berlin, 1842; and "Recherches sur diverses applications de l'analyse infinitesimale a la th6orie des nombres," Journal fur die reine und angewandte Mathematik, vol. XIX, p. 324, and vol. XXI, p. 1, Berlin, 1839, 1840. 2 Gauss and Dirichlet designate ax2 + 2bxy + cy2 by (a, b, c), but Kronecker designates ax2 + bxy + cy2 by the same symbol (a, b, c). The greatest common divisor of a, 2b, c is designated by o, and D = b2 - ac. 3 G. L. Dirichlet, "Zahlentheorie," 4th ed., ~ 54, p. 130, Braunschweig, 1894. 7 82 THE PELL EQUATION Substituting (4) in (3) (5) aXA + 2biv + cvp = 0. Combining (2) and (5) to eliminate first b, then c, we have a/. + cv = 0 a(X - p) + 2bv = 0, in which a 4 0, since otherwise D is a perfect square, which is contrary to hypothesis. Let au cu v=-' = -o' then 2bu (6) X- =- Substituting v and / in (1), (7) X ac2 + From (6) and (7), ~(X +~ p) 2 4(Du2 + l2) (X +q-p)2 = (X - p)2 + 4Xp = (- 2, 2 ]= Du2 + 2. Now let t = o-(X + p), which from the above equation must be an integer. Then 2- Du2 = E2 Q.E.D. The proof of the converse theorem could be given in a similar manner, and we see that we have a means of finding all1 the solutions of this type of indeterminate equation. Let us, for example, apply the theorem and the theory of reduced forms to finding the fundamental solution of x2 - 79y2 =. Here D = 79 and we take for this determinant the reduced form (7, 3, - 10). Then the greatest common divisor of 1 Dirichlet, "Zahlentheorie," p. 211. THE PELL EQUATION 83 7, 3, 10, is 1. The equation corresponding to this form is 7 + 2.3o - 10w2 = 0, from which -3 = 79 = -10 and the integral part of o, say kl, is 1. For the next form, (a', b', c'), make a' = - 10, the same as the last member of the preceding form. Then b' is obtained by the congruence b + b' 0 (mod- 10), 3 + b' O (mod- 10), whence b' -7 (mod - 10), and whence b' = 7. Furthermore c' may be obtained from the equation b'2 - a'c' = 79, whence 49 - (- 10)c' = 79, c' = 3. Then the new form is (-10, 7, 3). In a similar manner the remaining forms of the period are obtained; so that the entire period is (7, 3, -10), (-10, 7, 3), (3, 8, - 5), (- 5, 7, 6), (6, 5, - 9), (- 9, 4, 7). Furthermore, the integers k2, k3, k4, k5, k6, corresponding to kl of the first form are obtained at a glance. For example, add 8, the greatest integer in 4'79, to 7 and divide by 3, and the integer is 5. The k's are then 1, 5, 3, 2, 1, 1. This means that 3+ 79 1 1 + 5+ 10 5 + - - 1 3+ 1 1 2+ 1 1+1+... =(1,5,3,2,1, 1,...). 84 THE PELL EQUATION The successive convergents are 1 6 19 44 63 107 1' 5' 16' 37' 53' 90 From the last two we obtain the substitution (AX, u _ {53, 90 \ v, pJ 63, 107J from which comes the fundamental solution of the equation x2 - 79y2 = 1. If we wish to abridge the computation we need to build only the denominators up to, = 90 (or only the numerators to v = 63). From the formula - u we have, since in this case a = 1, u = 9, t = aI + Du2 = /1 + 7981 = 80, or, instead, t = X + bu = 53 + 3.9 = 80; and we might have taken still other reduced forms. If we begin with (1, 8, - 15) we obtain a period of four members, (1,8, - 15)(- 15, 157, 2), (2, 7, -15), (-15, 8,1). The k's are 1, 7, 1, 16. The convergents are 1 8 9 152 1' 7' 8' 135' of which the last two give the substitution (8, 135 \9, 152) and the resulting solution is 80, 9, as before. A new solution can be obtained from any two solutions, thus: (1) - t+ u f5D = (tl + ul D)(t2 + u2 /D), THE PELL EQUATION 85 When T, U, is the fundamental solution and n = 1, 2, 3, ~* co, the formula (2) (2) tn + un D = = (T + UlD) gives all the solutions without exception and without repetition.' Euler had already shown how from one solution an infinity of others may be obtained. He used formulas similar to the two given above. To show that the last formula contains all the solutions Dirichlet would proceed as follows: If (2) does not give all the solutions, then there is one, t, u, such that t is between tn and tn+l and therefore u between u, and u,+l where n > 1. Then tn + UnD < t + U D < tn,, + un+l4iD. Multiplying by tn- un /D, we have (ta, + un -ID) (tn, - u~, ~]D) < (t + u ~/D) (t U - u, ]D) < (T + U D)(tn + Un ID)(tn - U D), 1 < tt. - Duu,L + (Utn - tUn) ~D < T + U /D, and as (ttn - Duu,)2 - (utn - tun)2D = 1 there exists a positive solution less than the smallest, which is contrary to the hypothesis. As has been stated, Lagrange was the first to prove that x2 - Ay2 = 1 always has a solution when A is a positive non-square integer. This was one of his most significant contributions to the theory of numbers. The principles which he used were generalized and extended to higher problems by Dirichlet and others.2 The proof here given is independent of the theory of continued fractions and is 1Dirichlet, "Zahlentheorie," 4th ed., p. 212. 2 Monatsberichte der Berliner Akademie, October, 1841, April, 1842 March, 1846. Comptes rendus de la Academie, vol. X, p. 286, Paris, 1840. P. Bachman, "De unitatum complexarum theoria," 1864. 86 THE PELL EQUATION based on Dedekind's modification of Dirichlet'sl demonstration. I. Let y be an integer which varies from 0 to n so that 0 < y < n, and let A be a positive non-square integer and x the integer immediately superior to y IA, then 0 < x - yAIA < 1. (The equality symbol is needed only for the case y = 0, x = 1.) The value of any x - y lA is comprised between two of the fractions 0 1 2 n n n n n As there are only n intervals and as y, and therefore x - y /A, may take n + 1 values, there is at least one interval containing two values, and we may therefore write 0 < (Xi - yi AA) - (X2 -y2 AA) < or 0 < (xl - X2) - (Y1 - y2) 4A I < Now I y - y2 is one of the values of y, and letting xI - x2 be represented by x, we have 0 <x-y JA < < y. n y Now let m be an integer so large that the smallest value of x - y AIA shall be greater than 1/m. Then proceeding in the same manner, using m instead of n we get a new set, x, y, which gives a new solution to (1) 0 <x -y A <, Dirichlet, "Zahlentheorie," 4th ed., p. 372. A. Aubry, "Theorie de l'equation de Pell," Mathesis, vol. V (3), p. 233, Paris, 1905. THE PELL EQUATION 87 and so on repeatedly. We may then find an infinity of sets of values which satisfy (1). II. Starting from 0 <x - y WA <, and adding 2y /A, we have 0 < x +y A < + 2y'A. Multiplying together the two relations just given, we have 0 < x2 - Ay2 < + 2 A/A < 1 + 2 IA. Then if we designate by B some particular integer between 0 and 1 + 2 A/A, the equation x2 - Ay2 = B has an infinity of solutions. III. Among the infinity of solutions of the equation 2 - Ay2 = B there can not be more than B2 sets of values for x, y, such that, when x and y are divided by B, the remainders constitute all the combinations of numbers less than B. Therefore there are an infinity of sets which give the same remainders, and we may write with xl, yl, and x2, y2, two different sets of positive solutions, Xi2 - Ay12 = X22- Ay22 = B, x2 = xi+ aB, y2 = yi + fB; and if we let 1 + oax - AOyl = x, and ayi - f3x = y, then (xl - y o/A)(x2 + Y2 /A) = B(x - y 'A), and (Xi + yi WA)(x2 - Y2 AA) = B(x + y WA). 88 THE PELL EQUATION Multiplying these two equations and dividing by B2, we have x2 - Ay2 = 1, in which y $ 0; for if y = 0, then x = + 1, x1 -y A = (X2 -Y2 1A) and X1 = = x2, Y1 = - Y2, all of which are contrary to hypothesis. The theory of the equations x2 - Ay2 = 1 and X2 - A y2 = 4 is connected in a remarkable manner with that of the division of the circle. This was brought out by Dirichlet' and Jacobi.2 Dirichlet says in introducing his method: "It is not necessary to add that this mode of solution is much less suitable for numerical calculations than that developed from the use of continued fractions. This new manner of solving the equation t2 - pu2 = I ought to be considered only theoretically and as a bond of union between two branches of the science of numbers." It is to be observed that the solution obtained by these methods is not in general the least solution. Its ordinal place in the series of solutions depends on the number of classes of forms of the determinant of the form considered. Dirichlet's solution when p is an odd prime is as follows: Consider the equation p - 1 (1) = X = 0. G. L. Dirichlet, "Sur la maniere de resoudre l'6quation t2 - pu2 = 1 au moyen des fonctions circulaires," Journal fir die reine und angewandte Mathematik, vol. XVII, p. 286, Berlin, 1837; "Werke," vol. I, p. 345, Berlin, 1889. 2 C. G. J. Jacobi, "UJber die Kreistheilung und ihre Anwendung auf die Zahlentheorie," Journal fur die reine und angewandte Mathematik, vol. XXX, p. 166, Berlin, 1846. THE PELL EQUATION 89 The roots of this equation are given by the expression 27ri z/z e P, in which e and r have their ordinary meaning and m designates an integer of the series 1, 2,3,...p - 1. Among these integers there are (p - 1)/2 quadratic residues1 and also the same number of non-residues of p which we taken in a certain order and designate respectively by al, a2, *.. apl and bi, b2,.* bpi. 2 2 Then from the theory of Gauss we have the two equations 27Ti 27Tri 2r'i ~ al — a2- a(p-l)]2 Y+Z~a=p=2(x-e p (x-e px ). (x-e ), (2). b 2T ri b2 27ri,. 2-i, Y-Z p=2(xe )(x-e *... (x-e )/2 the upper or lower signs being taken according as p is of the form 4n + 1 or 4n + 3, and Y and Z being polynomials in x with integral coefficients. Multiplying these equations, (4) 4X = y2:= pZ2. As the numbers al, a2, a(p_l)/2, without regard to their order, are the remainders given by the squares 12, 22, * [(p - 1)/2]2 when divided by p, the first of the equations (2) can be replaced by 12 27ri 22 2w p12 2r (4) Y+Z = p==2(x-e P)(x-e P )...(x-e 2 ). Suppose first that p is of the form 4n + 1. Then let x = 1 in (3) and (4), and designate by g and h the corresponding integral values of Y and Z. Then (5) g2 - ph2 = 4p and ri 27 i 1 p-l\227ri 12. 22 2F. Gauss. 62 18 t g+ hp = 2(1-e )(1-e p).. (1 - e2 / p). 1 C. F. Gauss, "Disquisitiones," ~ 357, p. 637, Leipzig, 1801. 90 THE PELL EQUATION As we have s2. 2 27r 2. 1-e x = -2i.sins2-e P, p the last equation takes the form _ p+l p —1 - P+l P-l 7r 7r g + hp = 22 (-1) 4 sin 12.- sin 22 sin P — 2 1 [e12+22+...+(2 -1)2] 2 p But 12 22 ** [P- 1] P2 - 1 2 p 24 where (p2 - 1)/24 is an integer, odd or even according as p has the form 8n + 1 or 8n + 5. The exponential factor is either + 1 or - 1 and can be expressed as (- )-l)/ 4. Substituting, pw- I /_ l g+h/p-=2 2 *sin1. sin22-.. sin 2) p p 2 p From (5) we see that the integer g is divisible by p. Putting pk in place of g, h2- pk2 = - 4, and p+1 h + k1p= X sin12 sin22.sin 2 p p 2 p We see then that there exist integers h and k such that h2 - pk2 = - 4, and that these integers can be expressed in general by circular functions. From the preceding equations a 2 1 (a 2 =k= —[~ -. h 2 a k \ 2+ a/ After distinguishing the cases p = 8n + 1 THE PELL EQUATION 91 or p = 8n +5, we can from these values go back to those of t and u in t2 - pu2 = 1. Elliptic functions were first used in the solution of the Pell equations by Kronecker.1 By his method he shows in a surprising manner how several different algebraic expressions can be represented approximately by the same expression. Thus for D = 5, 13 or 37, the expressions 2 + 415, 18+5 413, 882 + 145 -,37, are all represented approximately by 1 D 8 ' It is interesting to note also the diverse forms which are obtained by elliptic functions for one and the same solution of the Pell equation. Thus in the solution of 2 - 17y2= - 1, for the expression 4 + 417, both of these approximate expressions are obtained: -2 e- ls 17 1 9 ' /5 Kronecker goes on to say, "The consequences which spring from the preceding results are very important. We not only discover a surprising relation between quadratic forms corresponding to opposed determinants, but, moreover we perceive here the decomposition of equations into singular moduli, which formed the principal object 1M. Kronecker, "Sur la resolution de l'equation de Pell au moyen des fonctions elliptiques," translated into French by M. Hotiel, Annales scientifiques de l'Ecole Normale Superieure, vol. III, p. 303, Paris, 1866, from the Monatsbericht der Akademie der Wissenschaft, p. 44, Jan. 22, 1863, Berlin. 92 THE PELL EQUATION of a note which I presented in June, 1862. This same decomposition has made easily recognizable the possibility of representing by means of singular moduli of elliptic functions certain solutions of the Pell equation." The importance of this equation will again be recognized in determining the units of a real quadratic domain, f( -Im). Finding the units, besides = 1, of such a domain, is equivalent to solving x2 - my2 = 1, if m 1 (mod 4), and (x+y) — m)2_ = - 1, if m 1 (mod 4), at least for the + sign in the right member, for every integral value of m. One method of attack for the solution of x2 - my2 = + 1 or of (x+ Y - (Y) =M 1 is to seek out first an ambiguous principal ideal which is different from (<em), and then from the square of this principal ideal to obtain the fundamental unit e. The solutions of the equations mentioned may always be deduced from the solutions of x2 - my2 = - a and where2 a where a is a factor of 4m or m. The latter values are always much smaller than those sought. This method fails when the domain f( /m) contains only the ambiguous principal ideal (]im). Even in this case shorter methods than the standard continued fraction procedure may be found.1 1 J. Sommer, "Vorlesungen iuber Zahlentheorie," p. 343, Leipzig, 1907. THE PELL EQUATION 93 If in the notation of hyperbolic functions, we put = cosh-' () sinh- ( I), we shall have Tn = - cosh no, Un ID = a sinh np. The convenience of this is that the known formulas of hyperbolic functions may be used to express the relations between the different values of Tn and U,, of the equation T2 - DUn2 = -2 Thus from the formulas, cosh 2sp = 2 cosh2 p - 1 sinh 2(p = 2 sinh -cosh cp, we deduce oT2n = 2Tn2 - 72, oU2n = 2TnUn, and so on in other cases. It may be specially observed that 2T 2T Tn+1 = T n -T 1, Unl1 = - Un - Un-1. These formulas are very convenient for calculating the successive values of T and U.1 Several classes of fundamental solutions of x2 -Ay2 = 1 have been noted in which the corresponding values of A, x, y, occur in arithmetical progressions. Thus, for example, one class2 would be A = 5, 10, 17, *, x = 9, 19, 33,., y 4, 6, 8,., in which A = m2 + 4m + 5, x = 2m2 + 8m + 9 (m = 0, 1, 2,* *), 1 G. B. Mathews, "Theory of Numbers," Part 1, p. 93, Cambridge, 1892. 2 G. Speckman, "Fundamental auflisungen der Pell'schen Gleichung," Archiv der Mathematik und Physik, vol. XIII (2), p. 327, Leipzig, 1895. 94 THE PELL EQUATION and the y's are the even numbers beginning with 4. Another class contains A = 7, 14, 23,.-, x = 8, 15, 24,, y = 3, 4, 5,., in which A = m2 + 6m + 7, x = m2 + 6m + 8 (m = 0, 1, 2, * ), and the y's are the natural series beginning with 3. Similar classes have been formed by Speckman1 for y = 1 and y = 2. But classes can be formed for every value of y. Thus for y = 5 we might have A = 25m2 + (50 =- 8 =- 6)m + 26 - 8 =- (6 - 1), x = 25m + 25 == 4 - 3, y = 5, in which m takes all integral values from 0 to o, and the double signs may be read in any one of four ways, all plus, all minus, alternately plus and minus beginning with either sign. For y any even number we have A = m2n2 + (4m2 - 1)n + 4m2 = 2, x = 2m2n + 4m2 = 1, y = 2m; and for y any integer, a general class would be A = m2n2 + (2m2 = 2)n + m = 2, x = m2n+m = 1, = m, in both of which classes m and n take all values from 0 to co. 1 Speckman, loc. cit., says that beyond y = 2 the formation of classes seems to be very difficult. THE PELL EQUATION 95 Euler calculated the fundamental solutions of x2 - Ay2 = 1 from A = 2 to A = 68 as published in the Commentaries1 of the St. Petersburg Academy, and to A = 99 as given in his algebra.2 In the Novi Commentarii his table occupies two pages and is entitled, "Tabula numerorum p et q quibus fit pp = Iqq + 1 pro omnibus valoribus numeri 1 usque ad 100." A line is drawn across all the columns to denote the omission of the square numbers. At the conclusion of the table solutions are given for the numbers, 103, 109, 113, 157, 367. In the first edition of Legendre's Theory of numbers,3 table XII at the end of the volume contains the solutions, written in the form of a ratio m/n, of m2 - an2 = - 1 when possible, otherwise of m2 - an2 = 1 for all non-square values of a from 2 to 1,003. There is no mark to indicate to which equation the solution, m, n, belongs. There are errors4 in one or both the values, m, n, under the following values of a: 133, 214, 236, 301, 307, 331, 343, 344, 355, 365, 397, 501, 526, 533, 613, 619, 629, 655, 664, 671, 694, 718, 732, 753, 771, 801, 806, 809, 851, 856, 865, 871, 878, 886, 944, 965, 995, 1,001. The most of these are corrected in the third edition. In Legendre's second edition, table X at the end of the volume contains the Pellian solutions with the same arrangement as before but carried only to the argument 135. The Canon Pellianus of Degen5 was published in 1817. 1 L. Euler, Commentarii Academiae scientiarum imperialis Petropolitanae, 1732-3, vol. VI, p. 175, St. Petersburg, 1738 —"Commentaxii arithmeticalvol. I, p. 4, St. Petersburg, 1849. 2 L. Euler, "Algebra," vol. II, p. 328, St. Petersburg, 1770; translated with additions by J. L. Lagrange, vol. II, p. 133, Lyons, 1774. 3A. M. Legendre, "Essai sur la theorie des nombres," Paris, 1798. 2d ed., Paris, 1808, 3d ed., Paris, 1830. 4A. Cunningham, "On high Pellian factorisations," Messenger of mathematics, vol. XXXV, p. 182, London, 1906. 5 C. F. Degen, "Canon Pellianus sive tabula simplicissimam aequationis celebratissimae y2 = ax2 + 1 solutionem pro singulis numeri dati valoribus 96 THE PELL EQUATION Table I contains the solutions of y2 - ax2 = 1, for all non-square values of a from a = 1 to a = 1,000. Table II contains solutions of y2 - ax2 = - 1, when possible, within the same range of values of a; but though the title of the table says that the solutions are given when possible, the solutions for y2 - ax2 = - 1 when a is of the form a 2 + 1 are omitted. In this case y = a, x = 1 would be a solution. It should also be noted that the usual symbols y and x are transposed. Besides the solutions the tables of Degen also give the elements of the continued fractions which lead to them, and this practice is continued in the tables in the British Association Reports. The following corrections1 should be made: a = 238, for x = 1,756 read x = 756;? =437, for y = 4,499 read y = 4,599; a = 672, for'y = 327 read y = 337; a = 751, the last seven figures of! "y uld be.. 4,418,960; a = 823, insert 47 after 235,170 in the value of y; a = 919, the last fourteen figures of x should be 36,759,781,499,589; a = 945, for y = 27,551 read y = 275,561; a = 951, for y = 22,420,806 read y = 224,208,076. In Legendre's third and fourth editions3 the table is nearly the same as in the first edition, but the values of x and y are printed in the form of a ratio x: y, and instead of being printed in one list they are printed in several according to the number of figures in the numbers x and y. This table also has the inconvenience of the former table in that there is no distinction between the solutions of x2- Ny2 = 1 and x2 - Ny2 = - 1. The following corrigenda have been noted: N = 94, read x = 2,143,295; N = 116, read x = 9,801; N = 149, read y = 9,305; N = 308, read x = 351; N = 479, read y = 136,591; N ab 1 usque ad 1000 in numeris rationalibus iisdemque integris exhibens," Copenhagen, 1817. 1 A. Cunningham, op. cit., p. 183. 2 C. F. Degen, op. cit., p. 112. 3 Legendre, "Theorie des nombres," 3d ed., Paris, 1830, 4th ed., Paris, 1900. THE PELL EQUATION 97 = 629, read x = 7,850; N = 667, read y = 4,147,668; N = 271, x should end with* * 983,600; N = 749, x should end with... 84,895; N = 751, x should end with... 424,418,960; N = 823, insert1 47 after 235,170 in x; N = 809, x should begin with 43,385* -..2 The British Association report3 for 1893 contains a table which continues the solution of one or the other of the equations y2 - ax2 = - 1 and y2 - ax2 = 1 from a = 1,001 up to a = 1,500, for all non-square values of a. The solution of the former equation is given when possible, of the latter in other cases, and there is the inconvenience of having the two solutions given at the same time. The only method of showing that the solution is for the equation y2 -x2 = - 1 is the placing of an asterisk (*) after the- argumnent, and even this is omitted under a = 1,361. Allan Ju.lningham4 has published tables of the fundamental solutions of r2 - Dv2 = 1 for all non-square values of D <4100, and for the same range, when the solution exists, for the equation r2 - Dv2 = - 1. He also gives tables of multiple solutions, To,vo; z1, v1; T2, V2;... of both the Pell equations, for all non-square values of D from 2 to 20. Barlow5 gave a table of solutions of p2 - Nq2 = 1 for every non-square value of N from 2 to 102. For other isolated solutions see the bibliography. 1 H. Richaud, Journal de mathematiques 6elmentaires, p. 183, vol. XI, Paris, 1887. 2 Richaud, loc. cit. For N = 629 see E. Catalan, Atti dell' Accademia pontificia de' Nuovi Lincei, vol. XX, p. 3, Rome, 1867; and C. A. Roberts also makes a correction, Mathematical magazine, vol. II, p. 105, Washington, 1892. 3 C. E. Bickmore, "Report of a committee appointed for the purpose of carrying on the tables connected with the Pellian equation from the point where the work was left by Degen in 1817" (text by Professor Cayley), British Association Report, vol. LXIII, p. 73, Nottingham, 1893, London, 1894. A. Cayley, "Collected mathematical papers," vol. XIII, p. 430, Cambridge, 1897. 4 A. Cunningham, " Quadratic partitions," London, 1904. 5 P. Barlow, "An elementary investigation of the theory of numbers," p. 507, London, 1811. 98 THE PELL EQUATION The present paper extends these solutions from the point where the British committee stops to A = 1,700 inclusive. It has been thought best not to omit the solution of the important equation x2 - Ay2 = 1 even in the case in which x2 - Ay2 = - 1 has a solution. In this respect the table presents an added convenience, different from former tables. The solutions have been made by first reducing A1I to a continued fraction, and then constructing tables for each number, as is shown in the two models given below which not only give the solution, x, y, of the equation x2 - Ay2 = 1 but also the solution, x, y, of the equation x2 - Ay2 = B for the numbers | B I < 2 iA, when a solution exists. In the first column are the indices of the development of 1A; in the second and third columns the values of x and y which substituted in the form x2 - Ay2 give the value named in the fourth column. In the first line x is always taken as 1, and y as 0; and in the second line x = the square root of the greatest square in A, and y = 1. Thereafter any value of x is obtained by multiplying the value of x on the line above by the index on the same line in the first column and adding to the product the value of x in the next line above. The same procedure gives the values of y. The fact that the middle of the period is reached in the development of the continued fraction is indicated by parentheses around the index and also, in the continued fraction table, around the number beneath it which denotes the corresponding value of the form x2 - Ay2. The two model tables differ in that in the second the period of the continued fraction contains an odd number of terms, and parentheses are placed around the two indices in the center to indicate this. That the number of terms in the period of -A is odd is a necessary and sufficient condition for the solution of x2 - Ay2 = - 1. In such cases both the solution of x2 - Ay2 = - 1 THE PELL EQUATION 99 and of x2 - Ay2 = 1 are given, the smaller set, of course, being the solution of x2 - Ay2 = - 1. In fact, it is worthy of notice that the fundamental solution of x2 - Ay2 = 1 occurs last and is precisely the most complicated of all the fundamental solutions of the form x2- Ay2 = B when such solutions exist at all. If -A is developed into a continued fraction, we have WA = k + 1 k2+ k+ k2 ------ kl +or, as it might be written, -A = (k, ki, k2, *., k2, ki, 2k, *.). Let us now take for two model examples A = 1,953 and A = 1,546. Then.1,953 = (44, 5, 5, 3, (12), 3, 5, 5, 88, *..), 1, 17, 16, 27, (7), 27, 16, 17, 1,..., Y1,546= (39, 3, 7, (1), (1), 7, 3, 78,.-.), 1, 25, 10, (39), (39), 10, 25, 1,..., and we construct tables in the manner described. 100 THE PELL EQUATION MODEL OF EXTENDED TABLE FOR 1,953. hk x y X2 - 1,953y2 44 1 0 + 1 5 44 1 -17 5 221 5 +16 3 1,149 26 -27 (12) 3,668 83 (+ 7) 3 45,165 1,022 -27 5 139,163 3,149 +16 5 740,980 16,767 -17 88 3,844,063 86,984 + 1 MODEL OF EXTENDED TABLE FOR 1,546. ki x y X2 - 1,546y2 39 1 0 +1 3 39 1 -25 7 118 3 +10 (1) 865 22 (-39) (1) 983 25 (+39) 7 1,148 47 -10 3 13,919 354 +25 78 43,605 1,109 - 1 The first table shows that 12 - 1,953.02 = + 1, 442 - 1,953.12= - 17, 2212 - 1,953.52 = + 16, and 3,844,0632 - 1,953-86,9842 = +1, giving the fundamental solution of x2 - 1,953y2 = 1. In practice, computation has been shortened by using the formula qzl,-l= q- A (qj+l + q,"1). Thus y = 83(1,022 + 26) = 86,984, and the value of x has been verified by substituting in the formula x = IAy2 + 1, so that it is believed that the solutions are correct, for the slightest mistake in the continued fraction or in THE PELL EQUATION 101 the extended table would lead to most erroneous results in the fundamental solutions. In a similar way when the period is odd, like the model in the second extended table, only half the numbers in third column need be computed. It has been the practice to use the formula q2 = q 2 + q+l12 Thus q2, = 222 + 252 = 484 + 625 = 1,109. After p2, has been found by, say, P2, = 4Aq,2 - 1 = 43,605, or by using the formula P2g = p,~+lq+l + pq., or P2, = kq2 + q2g-1, we see that x = 2p2,2 + 1 = 2(43,605)2 + 1 = 3,802,792,051, and y = 2p2, q2, = 2.43,605 1,109 = 96,715,890, so that 43,6052 - 1,546.1,1092 = - 1 and 3,802,792,0512 - 1,546-9,671,5802= + 1. If, however, in any problem we get p2 - Aq2 = 2, then x and y are computed by the formulas x = p2 =F 1, y = pq; and similar formulas are applied to p2 -Aq2= - 4. Making use of the table of continued fractions which is given in appendix A, the following fundamental solutions have been condensed from tables computed like the two model extended tables just described. The value of y is given before that of x. Where x2 - Ay2= -1 is solvable for example, for A = 1,514, the solution of x2- Ay2 = - 1 is given first and then the solution of x2- Ay2 = 1 directly after it, and the latter is further distinguished by the fact that it consists of much larger numbers. 102 THE PELL EQUATION TABLE OF SOLUTIONS 1501 349348930906018990606250760 13534735238019723614555987551 1502 153939320 5966017599 1503 1456 56447 1504 55266 2143295 1505 308508 11968361 1506 821517752 31880816705 1507 126063 4893778 1508 6 233 1509 1657656 64393055 1510 2163051048 84053391679 1511 989404834498003 38459732591163960 1512 225 8749 1513 572803224 22280499233 1514 1304305 50750707 132388801787270 5151268521999699 1515 13 506 1516 8602591618573944625072398763290 334949171001860160500111891352199 1517 760 29601 1518 26 1013 1519 39 1520 THE PELL EQUATION 103 1520 1 39 1522 1 39 78 3043 1523 39 1522 1524 26 1015 1525 761 29718 45230796 1766319049 1526 2614568 102135615 1527 13 508 1528 4221027300 164998439999 1529 1848616601300 72285400494999 1530 5876 229841 1531 29571423275371204812741659877 1157070504812569705163513222450 1532 17998512 704475647 1533 1684020 65935351 1534 6 235 1535 35175 1378124 1536 1176245 46099201 1537 1915090212 75080329577 1538 20746 813603 1539 500929 19651490 1540 2442 95831 104 THE PELLI EQUATION 1541 21203381993734892 832350906580405815 1542 66010 2592101 1543 169120149360850062054141 6643215129085261568619128 1544 2839 111555 1545 3141516 123481961 1546 1109 43605 96715890 3802792051 1547 3 118 1548 616314 24248647 1549 3940603592440589186792668219155493 155091664786017897306106048533964770 12223088243387474959342426833477559788180762631786819660 -98672827963220 481068489721970877435886874814139750846986322481107506339 -52591202305801 1550 25400 999999 1551 46980 1850201 1552 1594338 62809633 1553 2330233 91830104 427971077468464 16865536001301633 1554 4066 160285 1555 129784305853476 5117847907209209 1556 83330 3287049 1557 316747533960 12498490045601 1558 452808030 17873016101 THE PELL EQUATION 105 1559 53759147640581 2122635541940760 1560 2 79 1561 4338388198177894781607072 171407516109350655091252735 1562 45878 1813197 1563 608437685 24054459026 1564 2723626324440 107712448284799 1565 12226525 483682318 11827507906169900 467897169491706249 1566 309792110 12219743051 1567 66543 2634128 1568 495 19601 1569 92754893170353950236 3674077401859095458455 1570 302693 11993673 7260801722778 287696384061859 1571 36080080863 1430064397610 1572 1171222 46437143 1573 21240 842401 1574 78066512141574 3097184950960165 1575 51 2024 1576 7861342753913055 312086396361222451 1577 2828573966428 112326799954263 1578 11136 442367 106 THE PELL EQUATION 1579 21463242324201893177981952219891 852876982639022059460736831316490 1580 4 159 1581 7320852 291090185 1582 216060 8593649 1583 2370444310151 94312707829728 1584 5 199 1585 10021140809, 398962243872 7996113646631818744896 318341744070762403104769 1586 325 12943 8412950 335042499 1587 24553 978122 1588 21047119895869216830 838721786045010184649 1599 3182620572954663804 126866457849998708855 1590 8 319 1591 289337903407635 11540919718817324 1592 10 399 1593 1564 62423 1594 586466207477 23414622020115 27463769151288592914799710 1096489048689708486929226451 1595 16 639 1596 20 799 THE PELL EQUATION 107 1597 12755976753725984792525 509760496584162107935182 13004986088790772250309504643908671520836229100 519711527755463096224266385375638449943026746249 1598 40 1599 1599 1 40 1601 1 40 80 3201 1602 40 1601 1603 1811192895 72515603726 1604 20 801 1605 16 641 1606 612312261798 24538370918885 1607 3418785065885 137050219854024 1608 10 401 1609 5745796381280420432166845 230477350888785807136105668 2648551857407766364935405630459204820419664252354920 106239618545424991765243316655636841919665809723452449 1610 8 321 1611 104397 4190210 1612 22353158982 897473069767 1613 37 1486 109964 4416393 1614 7260775270 291698879051 1615 81372 3284569 108 THE PELL EQUATION 1616 5 201 1617 55568 2234497 1618 26737 1075479 57510164046 2313310158883 1619 1939851949127419161 78053433117061872410 1620 4 161 1621 1393793173905903098261469193463230841 56116404965454319198851772383057215250 156429324369979112128445583345098338627552043874824108399 -177922442751050500 6298101812493732343034974500091457815529942308667051412857 -352310169665125001 1622 70297763190 2831176412101 1623 1316 53017 1624 1475676 59468095 1625 13261 534568 14177812496 571525893249 1626 6612668183130 266647183921549 1627 363 14642 1628 66 2663 1629 61189099561153980 2469645423824185801 1630 373476558 15078465611 1631 975 39376 1632 505 20401 1633 8166047445137789303136 329993200416163472690687 THE PELL EQUATION 109 1634 611556. 24720785 1635 2852 115321 1636 619394999323609291080 25052977273092427986049 1637 2104652225 85153927318 358438805194633965100 14502382675358453346249 1638 36 1457 1639 34629 1401940 1640 2 81 1641 108' 4375 1642 931466115401 37744496576829 70315439208370501553286858 2849294043676512198L99390483 1643 221 8958 1644 121987690 4946145799 1645 233255566728 9460519131041 1646 1988 80655 1647 12 487 1648 2550475116 103537981567 1649 2680393 108845080 583495181032880 23694502880412801 1650 208 8449 1651 1382177919146568 56161343343007135 1652 2790 113399 110 THE PELL EQUATION 1653 35436240 1440734849 1654 366 14885 1655 7876 320409 1656 598 24335 1657 263494752020635645 10725887168686036632 5652429959428489976269630864789895280 230089311110767526480728245224891806849 1658 1073 43691 93760886 3817806963 1659 18356 747655 1660 16642394482596 678062702284831 1661 304653186299769440 12416252642220275199 1662 1578550 64353749 1663 4231222898258884355775 172548834009765974150624 1664 1326255 54100801 1665 4192880 171088001 1666 60 2449 1667 6665390367 272140632242 1668 2089134 85322647 1669 26478312606527663088656177 1081729001755127183954955210 57284717328038737412405689988022933413839259824966430 2340275266476287900450581841147662370297772626212288201 1670 2448162 100045691 1671 233835 9558676 THE PELL EQUATION 111 1672 828 33857 1673 77707680976 3178424641407 1674 41261000550 1688175067501 1675 11387618344557 466058366908226 1676 3566015383457670 145989028829001799 1677 840 34399 1678 238686492939208596 9777409878270240143 1679 41 1680 1680 1 41 1682 1 41 82 3363 1683 41 1682 1684 94536997975919510440249890353130 3879474045914926879468217167061449 1685 841 34522 58066004 2383536969 1686 860934422 35350768325 1687 843749566395 34655415003224 1688 170925 7022501 1689 782665679312412462996 32165559846732699944855 1690 666 27379 1691 814679883 33501079090 1692 112 4607 112 THE PELL EQUATION 1693 31441289541598765 1293685853230772118 81350302974597900525875082210468540 3347246173698861715400298156892411849 1694 293564 12082575 1695 6063 249616 1696 49978955055195 2058259344325201 1697 285252185 11750866732 6703920821893618840 276165737906448719649 1698 120070380 4947715601 1699 66555468361862588059573967039601 2743345037851573163040031857578450 1700 3484 143649 THE PELL EQUATION 113 BIBLIOGRAPHY. P. COSALI, "Origine, trasporto in Italia, primi progressi in essa dell' algebra," vol. I, p. 146, Parma, 1798. The work of Lagrange and Euler upon the equation AY2 + B = Z2 and its connection with the general indeterminate equation of the second degree in two unknowns is discussed. F. PEZZI, "Nuovi teoremi sulla posibilita' dell' equazione x2 - Ay2 = -= 1 e ricera del numero de' termini del periodo della radice quadra di un numero non quadrato, sviluppata in frazione continua," Memorie di matematica e di fisica della Societa Italiana delle Scienze, vol. XIII, p. 342, Modena, 1807. The author proves that x2 - Ay2 = 1 is always solvable, that x2- Ay2 = - 1 is solvable in integers if the number of terms in the period of the continued fraction in the development of i/A is odd, impossible if the number of terms is even. There are four theorems about the results in solving the equation Mn2 = ANn2 + ( - )n according as A,, N,, M,, are odd or even. Complete examples are given for A = 94 and A = 1,005. KRAMP, "Recherches sur les fractions continues periodiques," Annales de mathematiques pures et appliquees, vol. I, p. 261, Nimes, 1810 and 1811. Application is made to the equation lly2 + 49 = x2 (p. 283); and this particular equation is treated more fully by Dr. Kramp in another communication in the same journal (p. 319). TEDENAT, "Communiquee aux redacterus des Annales sur la lettre de M. Kramp," Annales de mathematiques pures et appliquees, vol. I, p. 349, Nimes, 1810 and 1811. This article gives the formulas for the solution of the equation y2 - Ax2 = B, obtained from the integration 9 114 THE PELL EQUATION of an equation of finite differences of the form y" -2my' + y = 0. P. BARLOW, "An elementary investigation of the theory of numbers," p. 294, London, 1811. There are fifteen theorems on the Pell equation; and the fundamental solutions of x2 - Ny2 = 1 from N = 2 to N = 102 are given. P. BARLOW, "New mathematical tables," p. 266, London, 1814. General formulas are given for the solution of the equations x2- ay2 = z2, x2 Ny2 = 1, x y2Ny = A. G. PALETTI, "Risilutione dell' equazione generale completa di secondo grado a tre indeterminate," Rome, 1820. The general solution of the indeterminate equation of the second degree in three unknowns is made to depend on the Pell equation. E. F. A. MINDING, "Observatio pertinens ad solutionem aequationum indeterminatarum secundi gradus," Journal fur die reine und angewandte Mathematik, vol. VII, p. 140, Berlin, 1831. P. N. C. EGEN, "Handbuch der allgemeinen Arithmetik," Part I, p. 457, Berlin, 1833, Part II, p. 467, Berlin, 1834. Egen gives the 121 values of A < 1,000 for which x2 - Ay2 = - 1 is solvable. STERN, "Theorie der Kettenbruiche und ihre Anwendung," Journal fiir die reine und angewandte Mathematik, vol. XI, p. 277, Berlin, 1834. This is one of seven lengthy articles on this subject. The applications to the Pell equation are contained on pages 327-341. There is a table of continued fractions for forty-two general forms. THE PELL EQUATION 115 G. L. DIRICHLET, "Recherches sur diverses applications de l'analyse infinitesimale a la theorie des nombres," Journal ffir die reine und angewandte Mathematik, vol. XIX, p. 324, vol. XXI, p. 1, Berlin, 1839, 1840. C. D'ANDREA, "Trattato elementare di aritmetica e d'algebra," vol. II, p. 671, Naples, 1840. That t2 - Au2 = 1 is always solvable in integers is proved through continued fractions. C. G. J. JACOBI, "Uber die Kreistheilung und ihre Anwendung auf die Zahlentheorie," Journal fur die reine und angewandte Mathematik, vol. XXX, p. 166, Berlin, 1846. If p is a prime of the form 4n + 1 and a designates the quadratic residues between 0 and ~p, then P-l ar (x + y l-p) /p = 22 I sin2 - where x2 - py2 = - 4, and II designates the product of the collected factors sin2 air/p; and if q is a prime of the form of 8n + 3 x+yq =. 2.nsin( + ) where x2 - qy2 = - 2. There is a table of the separations of the primes p of the form 4n + 1 into the sum of two squares up to p = 11,981. This article was first published in Monatsberichte der K6niglichen Akademie der Wissenschaften, 1837. F. ARNDT, "Bemerkungen iber die Verwandlung der irrationalen Quadratwurzel in einen Kettenbruch," Journal fur die reine und angewandte Mathematik, vol. XXXI, p. 343, Berlin, 1846. The author discusses p2 - Aq2 = 1, when A is an odd number, when A is an odd power of 2, and when A = 2A' where A' is odd. The work of Legendre in his "Theorie des nombres" is generalized and extended. 116 THE PELL EQUATION L. WANTZEL, " Divisors of numbers of the form x2 - cy2, Extraits des Proces-Verbaux des seances de la Societe Philomathique, p. 19, Paris, 1848. J. B. LUCE, "On the theory of numbers," American Journal of science and arts, vol. VIII, p. 55, New Haven, 1849. Application is made to the solution of p2 - nq2 = 1 for many special values of n. A table is given for reducing the numbers n up to n = 158 when considered "complex" for purposes of the solution, to "simpler" numbers by multiplying by a square factor. F. ARNDT, "Untersuchungen fiber einige unbestimmte Gleichungen zweiten Grades, und fiber die Verwandlung der Quadratwurzel aus einem Bruche in einen Kettenbruch," Archiv der Mathematik und Physik, vol. XII, p. 211, Grtfswald, 1849. This article closes with a table of the fundamental solutions of p02 - p''2 = 1 and p92 - pl02 = 2 from pp' = 3 to pp' = 1,003. P. TCHEBICHEFF, "Sur les formes quadratiques," Journal de mathematiques pures et appliquees, vol. XVI, p. 257, Paris, 1851. The whole treatise is a discussion of the equations x2 -Dy2 = N. P. VOLPICELLI, "Alcune ricerche relative alla teorica dei numeri, "Atti dell' Accademia pontificia Nuovi Lincei, vol. VI, p. 77, Rome, 1852. A. G6PEL, "De aequationibus secundi gradus indeterminatis," Journal fiir die reine und angewandte Mathematik, vol. XLV, p. 1, Berlin, 1853. The general equations x2 - Ay2 = i C are discussed, with such questions as for what values of A is the equation 2 - Ay2= + 2 solvable. THE PELL EQUATION 117 F. LANDRY, "Cinquieme memoire sur la theorie des nombres," Paris, 1856. This article contains a theorem in relation to the solution of x2 - Ay2 = rm when A + r = a2. There are applications to the methods of Lagrange and Gauss. M. A. STERN, "Zur Theorie der periodischen Kettenbriiche," Journal fiur die reine und angewandte Mathematik, vol. LIII, p. 1, Berlin, 1857. This article contains a hundred pages on the solution of the Pell equation by the aid of continued fractions, closing with what the author calls a Pellian table in which the numbers A(< 1,000) are classified. A. CAYLEY, "Note sur l'6quation x2 - Dy2 = - 4, D - 5 (mod 8)," Journal fur die reine und angewandte Mathematik, vol. LIII, p. 369, Berlin, 1857. The fundamental solution of the equation x2 - Dy2 = 4 is deduced from the fundamental solution of the equation t2- Du2 = -4 by means of x + y-JD =~ (t + u /D)2, giving x = t2 + 2, y = tu. The fundamental solution of the equation x2 - Dy2 = 1 is obtained from the fundamental solution of the equation T2 - DU2 = 4 by means of x +y D= I(T + UD)3, giving x =(T3 - 3T), y = (T2 -1) U. A table of solutions of the equation x2 - Dy2 = =- 4, D _ 5 (mod 8) is given for D = 5 to D = 997. Where such solutions are possible they are much smaller than for the equation x2 - Dy2 = 1. G. C. GERONO, "Resoudre en nombres entiers l'6quation x2- ny2 = 1, dans laquelle on suppose que n represente un nombre entier, positif, non carre," Nouvelles annales de mathematiques, vol. XVII, p. 122, 153, Paris, 1859. J. F. KONIG, "Zerlegung der Gleichung x2 - fgy2 = = 1 in Faktoren," Archiv der Mathematik und Physik, vol. XXXIII, p. 1, Griefswald, 1859. 118 THE PELL EQUATION Tables of such factors are given for various values of f and g up to f = 17, g = 89. H. J. S. SMITH, "Report on the theory of numbers," British Association report, p. 292, London, 1861; "Collected Mathematical papers," vol. I, p. 163, Oxford, 1894. This treatise contains many theorems relating to the Pell equations T2 - DU2 = 1 and T2 - DU2 = 4. C. RICHAUD, "Enonc6s de quelques theoremes sur la possibilite de l'equation x2 - Ny2 = - 1 en nombres entiers," Journal de mathematiques pures et appliquees, vol. IX (2), p. 384, Paris, 1864. C. RICHAUD, "Demonstrations de quelques theoremes concernant la resolution en nombres entiers de l'equation x2- Ny2 = - 1," Journal de mathematiques pures et appliquees, vol. X (2), p. 235, Paris, 1865. C. RICHAUD, "Sur la resolution des equations 2- Ay2 = 2,1," Atti dell' Accademia pontificia de' Nuovi Lincei, vol. XIX, p. 177, Rome, 1865. C. RICHAUD, "Sur l'equation x2 - Ny2 = -," Journal de mathematiques pures et appliquees, vol. XI (2), p. 145, Paris, 1866. These articles of Richaud contain many theorems on the solvability of the equation x2 - Ny2 = - 1, depending on whether or not the different factors of N are quadratic residues of each other. E. CATALAN, "Rectification et addition a la note sur un probleme d'analyse indeterminee," published in Atti dell' Accademia pontificia de' Nuovi Lincei, vol. XX, p. 1, Rome, 1867, Atti dell' Accademia pontificia de' Nuovi Lincei, vol. XX, p. 77, Rome, 1867. This note is upon the equation Ax2 - By2 = 1. THE PELL EQUATION 119 M. A. STERN, "Uber die Eigenschaften der periodischen negativen Kettenbriiche welche die Quadratwurzel aus einen ganzen positiven Zahl darstellen," Abhandlungen der Konigliche Gesellschaft der Wissenschaften, vol. XII, p. 3, Gottingen, 1866. In the continued fractions - 1 is used for the numerators. J. FRISCHAUF, in Situngsberichte der mathematischnaturwissenschaftlichen Classe der Kaiserlichen Akademie der Wissenschaften, vol. LV (Abtheilung 2), p. 121, Vienna, 1867. P. SEELING, "Uber die Formen der Zahlen, deren Quadratwurzeln in Kettenbriichen dargestellt, Perioden von einer gewissen Anzahl Stellen haben," Archiv der Mathematik und Physik, vol. XLIX, p. 4, Griefswald, 1869. This article contains tables for continued fractions with periods of from one to seven numbers, and for continued fractions for /IA for all values of A up to A = 602. It also discusses the forms of A for which x2 - Ay2 = 1. L. OTTINGER, "Uber das Pell'sche Problem und einige damit zusammenhangende Probleme aus der Zahlenlehre," Archiv der Mathematik und Physik, vol. XLIX, p. 193, Griefswald, 1869. Formulas for the general solutions of the equations x2-Ay2 = - 1 are given. If x2 - Ay2 = ==b is solvable, and if p, q, is one solution, and t, u, is a solution of the Pell equation so that p2 - Aq2 = 2 b and t2 - Au2 = 1, then x = pt = Aqu and y = pu - qt. Tables of solutions 120 THE PELL EQUATION of the equations x2 - Ay2 = b are given for A = 2, 3, 5,... 20, b = 1, 2,... 10, giving several solutions for each A and b; and for powers of b as far as x2 _ Ay2 = 7r+.1 A. LAISANT, MORET-BLANC, "Trouver un entier tel, que son carre augmente de 1 soit equal au double d'un carre," Nouvelles annales de mathematiques, vol. XXVIII, [VIII (2)], p. 336, vol. XXXI, [XI (2)], p. 173, Paris, 1869, 1872. N. DE KHANIKOF, "Procede pour resoudre, en nombres entiers, l'equation indeterminee A + Bt2 = u2," Comptes rendus de i'Academie, vol. LXIX, p. 185, Paris, 1869. L. CALZOLARI, "Nota sull'equazione Ax2 =- By2 = u2, Giornale di matematiche, vol. VIII, p. 28, Naples, 1870. Such theorems are proved as a necessary and sufficient condition for the solution in integers of the equation u2 = Ax2 = By2 is that the trinomial Ab2 - Ba2 =- AB be a square. P. SEELING, "fUber die Auflosung der Gleichung x2 - Ay2 = -= 1 in ganzen Zahlen, wo A positiv und kein vollstandiges Quadrat sein muss," Archiv der Mathematik und Physik, vol. LII, p. 40, Griefswald, 1870. At the close of this article there is a table of numbers A to A = 7,000 for which /IA has an odd period, and therefore for which x2 - Ay2 = - 1 is solvable. Thus x2 - 6,997y2 = - 1 has a solution. A. B. EVANS, A. MARTIN, "To find the smallest integer y which satisfies the relations 940,751y2 + 1 = o, and 940,751y2 + 38 = o," Mathematical questions from the Educational Times, vol. XVI, p. 34, London, 1872. There is no solution in integers to the equation 940,751y2 + 38 = x2. THE PELL EQUATION 121 The first part of this problem is said to have been solved by a student of Professor C. Gill at St. Paul's College, Flushing, about 1842, and at that time the numbers were the largest of the kind that had been found. x has 57 figures. O. SCHL6MILCH, "Uber die Kettenbruchentwickelungen fir Quadratwurzeln," Zeitschrift fir Mathematik und Physik, vol. XVII, p. 70, Leipzig, 1872. F. DIDON, C. MOREAU, "Solution of a question," Nouvelles annales de mathematiques, vol. XI (2), p. 48, vol. XII (2), p. 330, Paris, 1872, 1873. The indeterminate equation t2 - Du2 = 4, in which D is of the form (4n + 2)2 + 1, n designating any positive integer, 1, 2, 3,..., has no solution formed of two odd numbers, and the solution which is made of the two smallest positive integers is t = 16(2n + 1)2 + 2, u = 8(2n + 1). L. MATTHIESSEN, "Allgemeine Auflisung der Gleichung ax2 -* 1 = y2 in ganzen Zahlen," Zeitschrift fir Mathematik und Physik, vol. XVIII, p. 426, Leipzig, 1873, B. MINNIGERODE, "Uber eine neue Methode die Pell'sche Gleichung aufzul6sen," Nachrichten von der K6nigliche Gesellschaft der Wissenschaften, No. 23, p. 619, Gottingen, 1873. Reduced forms and continued fractions are used. G. H. HOPKINS, HART, "If the sum of the squares of two consecutive integers be equal to the square of another integer, find their integral values, and show how to find any number of particular solutions," Mathematical questions from the Educational Times, vol. XX, p. 63, London, 1874. Dr. Hart gives as the general solution of p2 - Nq2 = = 1, p = 2mr + r', q = 2ms + s' 122 THE PELL EQUATION in which m is the fundamental solution for p, r, and s the last found values for p and q, and r' and s' the next preceding values. Dr. Hart claims that this method after two solutions have been obtained is the simplest that has been yet devised for the succeeding solutions. W. SCHMIDT, "Uber die Auflisung der Gleichung t2 - Du2 = - 4, wo D eine positive ungerade Zahl und kein Quadrat ist," Zeitschrift ffir Mathematik und Physik, vol. XIX, p. 92, Leipzig, 1874. The solution is based upon the theory of quadratic forms. The example given is 2 - 61y2 = 4. M. COLLINS, A. M. NASH, "Prove that the equation x2 + Dm = (N2 + D)y2 is always possible in rational numbers for x and y when N and D are rational, and m is an odd integer, and that x and y can be found in integers when N and D are integers," Mathematical questions from the Educational Times, vol. XXII, p. 23, London, 1875. x= = NDn, y = D D, where m = 2n + 1. S. TEBAY, "Solution of a question," Mathematical questions from the Educational Times, vol. XXIII, p. 30, London, 1875. In order that nt2 = k be a square it is necessary that t = a + - n- (++ r r) {an)(qr -- - -- ) m(fqr + 7-r)} where v = p + qn', p, q, the smallest solution of X2 - ny2 = 1, and a any particular value of t. THE PELL EQUATION 123 S. BILLS, "A new method of solving in integers the equation x2 - Ay2 = 1= I," Mathematical questions from the Educational Times, vol. XXIII, p. 98, London, 1875. BOOTH, "Find the law which gives the value of t so as to make nt2 =t k a square," Mathematical questions from the Educational Times, vol. XXIII, p. 99, London, 1875. A. B. EVANS, HART, "Find the least integral values of x, y, that will satisfy the equation x2 - 953y2 = = 1,", Mathematical questions from the Educational Times, vol. XXIII, p. 107, London, 1875. A. MARTIN, "Correction of an error in Barlow's Theory of numbers," Analyst, vol. II, p. 140, Des Moines, 1875. S. BILLS, A. MARTIN, G. HART, A. B. EVANS, "Solution of a question," Mathematical questions from the Educational Times, vol. XXIII, p. 109, vol. XXIV, 9, 109, London, 1875, 1876. If R is an integral value of y which makes Ay2 + 1 a square and r is the smallest integral value of y which makes Ay2 - 1 a square, then R is a multiple of r. A. MARTIN, HART, "Find the least integral values of x and y that will make x2 - 5,693y2 = - 1," Mathematical questions from the Educational Times, vol. XXV, p. 97, London, 1876. A. MARTIN, "On the equation x2 - 5,658y2 = 1 in Barlow's Theory of numbers, p. 299," Mathematical questions from the Educational Times, vol. XXVI, p. 87, London, 1876. H. J. S. SMITH, "Notes on the theory of the Pellian equation and of binary quadratic forms of a positive determinant," Proceedings of the London Mathematical Society, vol. VII, p. 196, London, 1876, "Collected papers," vol. II, p. 148, Oxford, 1894, "Collectanea Mathematica," p. 117, Milan, 1881. This article contains the theorem that T + U - is 124 THE PELL EQUATION equal to the product of the complete quotients in the development of AID, theorems concerning the number of different periods of complete quotients and the number of non-equivalent classes of quadratic forms, and discussion of the nature of the periods. A. KUNERTH, "Neue Methoden zur auflosung unbestimmter quadratischer Gleichungen in ganzen Zahlen," Situngsberichte der kaiserlichen Akademie der Wissenschaften, mathematisch-naturwissenschaftliche Classe, vol. LXXV (Abtheilung 2), p. 7, Vienna, 1877. A. MARTIN, "Find the least integral values of x and y that will satisfy the equation x2 -- 9,817y2 = 1," Analyst, vol. IV, p. 154, Des Moines, 1877. The fundamental solution for x has 97 figures. D. S. HART, "Solution of an indeterminate problem," Analyst, vol. V, p. 118, Des Moines, 1878. General values of x, y, are found which satisfy the equation x2 - Ay2 = - 1 If A = r2 + s2, then y = - (m2 + n2) where m and n are determined from rn - l(r2 + s)n s 7n — In this value of m, r and s may represent any two numbers, one of which is even and the other any odd number except 1; n may be found by trial or by the solution of the equation p2 - (r2 + s2)n2 = S. E. DE JONQUIERES, "Decomposition du carre d'un nombre N et de ce nombre lui-meme en sommes quadriques THE PELL EQUATION 125 de la forme x2 - ty2, t etant un nombre rationnel, positif ou negatif; resolution en nombres entiers du systeme des equations indeterminees y = X2 + t(x + a)2, y2 = z2 + t(z + )2," Nouvelles annales de mathematiques, vol. XVII (2), p. 419, p. 433, Paris, 1878. HART, "A new method of solving equations of the form x2 - Ay2 = 1," Mathematical questions from the Educational Times, vol. XXVIII, p. 29, London, 1878. Let A = r2 = m, then x2 - r2y2 = 1 my2. Put x + ry = 1 - my2 and x - ry = 1; then 2 =~ my2 - my2 x= 2, ry= 2 Therefore 2r 2r m Y - m' x = m. Hence this method does not give integral solutions for all values of A. H. BROCARD, "Note sur le probleme de Pell," Nouvelle correspondance mathematique, vol. IV, p. 161, 193, 228, 337, Paris, 1878. S. ROBERTS, "On forms of numbers determined by continued fractions," Proceedings of the London Mathematical Society, vol. X, p. 29. London, 1878. Among the many theorems in this article the most interesting is: If t, u, is the least solution of x2 - Ay2 = 1, 126 THE PEIL EQUATION we shall have (except for M = 1) for some values t, u', less than t, u, either Mt'2 - N'2 = 1; MN = A; or Mt,2 -_N 2 = 2, MN = A. K. E. HOFFMAN, "Uber die Kettenbruchentwickelung fur die Irrationale 2. Grades," Archiv der Mathematik und Physik, vol. LXIV, p. 1, Leipzig, 1879. The author applies continued fractions to the Pell equation. A. KUNERTH, "Praktische Methode zur numerischen Auflosung unbestimmter quadratischer Gleichungen in rationalen Zahlen," Situngsberichte der kaiserlichen Akademie der Wissenschaften, mathematisch-naturwissenschaftliche Classe, vol. LXXVIII (Abtheilung 2), p. 327, Vienna, 1879. S. ROBERTS, C. LEUDESDORF, EVANS, "Show that the triangular numbers which are also squares are given by J (I + -2)2 _ (1 - +2 )2m 2 4 ~2 Mathematical questions from the Educational Times, vol. XXX, p. 37, London, 1879. By decomposing the expression we can obtain remarkable laws of derivation of the successive squares, 12.12, (1 + 1)2(2.1 + 1)2, (2.2 + 3)2(2 + 3)2 1, 22.32, 72.52, (7 + 5)2(2.5 + 7)2, (2.12 + 17)2(12+17)2, 122.172, 412.292, *.. EVANS, J. W. SHARPE, NASH, "If pn/qn be the last convergent in the first period of Al expanded as a continued fraction, and r the greatest integer contained in THE PELL EQUATION 127 A7, show that pn = rqn - qn-, " Mathematical questions from the Educational Times, vol. XXX, p. 49, London, 1879. This theorem gives a ready method of finding x, when y has been found in the equation x2- Ay2 = 1. L. RODET, "Sur une methode d'approximation des racines carrees connue dans l'Inde anterieurement a la conquete d'Alexandre," Bulletin de la Societe Mathematique de France, vol. VII, p. 98, Paris, 1879. The author thinks that the rule of Baudhayana is the foundation of Newton's method. T. PEPIN, "Sur quelques equations indeterminees du second degre et du quatrieme," Atti della Accademia pontificia de' Nuovi Lincei, vol. XXXII, p. 79, Rome, 1879. C. HENRY, " Sur une valeur approchee de 2 et deux approximations de 3d3," Bulletin des sciences mathematiques, vol. III (2), p. 515, Paris, 1879. LIONET, F. PISANI, "A question on the solution of x2 + 1 = 2y2," Nouvelles annales de mathematiques, vol. XVIII (2), p. 528, vol. XX (2), p. 373, Paris, 1879, 1881. The question concerns the law of recurrence in the solution; for example Xn = 6Xn-l - Xn-2. S. REALIS, "Sur quelques questions se rattachant au probleme de Pell," Nouvelle correspondance mathematiques, vol. VI, p. 306, p. 342, Paris, 1880. These articles give a method of deducing the chief formulas which relate to this problem and show that these formulas are capable of extension. 128 THE PELL EQUATION S. ROBERTS, "Notes on a problem of Fibonacci's," Proceedings of the London Mathematical Society, vol. XI, p. 35, London, 1880. This problem is to find x, y, v, u, for a given P in the equations x2 + Py2 = u2 and x2 - Py2 = v2. S. ROBERTS, "Note on the integral solution of 2 - 2Py2 = - 2 or - 2z2 in certain cases," Proceedings of the London Mathematical Society, vol. XI, p. 83, London, 1880. A. KUNERTH, "Berechnung der ganzzahligen Wurzeln unbestimmter quadratischer Gleichungen mit zwei unbekannten aus den fur letztere gefunden Briichen, nebst den Kriterien der Unm6glichkeit einer solchen Losung," Situngsberichte der kaiserlichen Akademie der Wissenschaften, mathematisch-naturwissenschaftliche Classe, vol. XXXII (Abtheilung 2), p. 342, Vienna, 1880. P. TANNERY, "Sur le probleme des boeufs d'Archimede," Bulletin des sciences mathematiques, vol. V (2), p. 25, Paris, 1881. P. TANNERY, " L'arithmetique des Grecs dans Pappus," Memoires de la Societe des Sciences physiques et naturelles de Bordeaux, vol. III (2), p. 351, Paris, 1881. Plato considered the equations 2y2 - x2 = - 1; Archimedes in his "Circle Measure" the equations 3y2 x2 = -1 and 3y2 x2 = 2. Diophantus was essentially influenced by Heron and Hypsicles. New conclusions are drawn from the "Cattle Problem" of Archimedes making it clear that an indeterminate analysis of no inconsiderable extent existed before the time of Christ. P. TANNERY, "L'arithmetique des Grecs dans Heron d'Alexandrie," Memoires de la Societe des Sciences THE PELL EQUATION 129 physiques et naturelles de Bordeaux, vol. IV (2), p. 161, Paris, 1882. W. DURFEE, " Notes on some properties of the numerical solutions of ax2 - y2 = - 1," Johns Hopkins University circular, vol. II, p. 178, Baltimore, 1882. If we have the integral solutions arranged according to magnitude designated by xi, yl; X2, Y2;... then XnYn+t - X = Xn+n Xt, aXnXn+t - YnYn+t = t. T. PEPIN, "The equations 2y2 - 1 = x, 2z2 - =x2 Fermat's theorem," Atti dell' Accademia pontificia dei Nuovi Lincei, vol. XXXVI, p. 23, Rome, 1883. E. C. CATALAN, "Sur l'quation Ax2 = y2 + 1," Association frangaise pour i'avancement des sciences, compte rendu, vol. XII, p. 101, Paris, 1883. This article gives the general solution, and discusses the decomposition of xn into the sum of three squares and of four squares. H. WEISSENBORN, "Bemerkungen zu den Archimedischen Naherungswerthen der irrationalen Quadratwurzeln," Zeitschrift fur Mathematik und Physik, historischliterarische Abteilung, vol. XXVIII, p. 81, Leipzig, 1883. The course of reasoning is based on a generalization of the side- and diagonal numbers of Theon of Smyrna. E. DE JONQUIERES, "Etude sur les fractions continues periodiques," Comptes rendus hebdomadaires des seances de l'Academie Sciences, vol. XCVI, p. 568, 694, 832, 1020, 1129, 1210, 1297, 1351, 1420, 1490, 1571, 1721, Paris, 1883. These articles have various headings, but the above title denotes the character of all. Theorems are established for determining, in many cases, from the number E the number of terms in the period of the continued fraction equal to 4E. The author's forms are compared to those of Lagrange. In some cases other numerators than unity are used. 10 130 THE PELL EQUATION E. PICARD, "Sur les formes quadratiques binaires a indeterminees conjugees," Comptes rendus de l'Academie, vol. XCVI, p. 1567, Paris, 1883. This article gives a general method of solving a generalized Pellian equation xxo - Dyyo = 1. DE ROQUIGNY, JAMET, BROCARD, EVEN, "Quels sont les polygons dont le nombre des diagonales est un carre?" Mathesis, vol. III, p. 216, vol. VI, p. 162, Paris, 1883, 1886. This is solved by the aid of the Pell equation (2v - 1)2 - 8u2 = 1. The results are 6, 27, 150, 867, *. E. MAHLER, "Die Irrationalitaten der Rabbinen," Zeitschrift fir Mathematik und Physik, historischliterarische Abteilung, vol. XXIX, p. 41, Leipzig, 1884. M. D'OCAGNE, "Sur l'6quation indeterminee x2 _ ky2 = z2, Comptes rendus de l'Academie, vol. XCIX, p. 1112, Paris, 1884. The solutions are stated by the aid of a certain function s(oa, 0, n), and are given without proof. H. WEISSENBORN, "Die irrationalen Quadratwurzeln bei Archimedes und Heron," Berlin, 1884. S. ROBERTS, "Notes on the Pellian equation," Proceedings of the London Mathematical Society, vol. XV, p. 247, London, 1884. The author shows that if we take the nearest integer limit in the development of a continued fraction a form will be arrived at whose coefficient is unity. This method sometimes gives a shorter solution than the usual method of always taking the inferior limit. The use of superior limits alone may prolong the operation, and does not THE PELL EQUATION 131 directly furnish the solution of the equation x2 - Ay2 = - 1, when possible. The case in which the first transformation is by an inferior limit and succeeding ones by superior limits is discussed. The author, before he wrote his article, did not know of the papers of M. A. Stern and B. Minnigerode. This is an example of how valuable the work of the historian of mathematics could be. WEILL, "Sur quelques equations indeterminees," Nouvelles annales de mathematiques, vol. IV (3), p. 189, Paris, 1885. General formulas for the solutions of the equations X2 - Ay2 = N2 and 2 - Ay2 = 1 are discussed. H. VAN AUBEL, "Quelques notes sur le probleme de Pell," Association frangaise pour l'avancement des sciences, compte rendu, part 2, p. 135, Paris, 1885. This article gives the general form for the solution in terms of the convergents found near the middle of the period of the continued fraction. It gives methods of finding the solution of the equation 2 - Ay2 = 1 quickly for many particular values of A. G. FRATTINI, "Intorno ad un teorema di Lagrange," Atti della Reale Accademia dei Lincei, rendiconti, vol. I, p. 136, Rome, 1885. There are many theorems upon the solvability of the congruence, x2 -Dy2 X (mod p). H. RICHAUD, "Solution of the equation y2 - 1549x2 = - 1," 132 THE PELL EQUATION Journal de mathematiques 6elmentaires, vol. XI, p. 182, Paris, 1887. A. MEYER, "Uber eine Eigenschaft der Pell'schen Gleichung," Vierteljahrschrift der Naturforschenden Gesellschaft in Zirich, vol. XXXII, p. 363, Zurich, 1887. The author makes use of indefinite ternary quadratic forms and proves this theorem: If D is a positive integer, 2a the highest power of 2 in D, a - 4, S2 the largest odd square in D and D = 20S2D1, then there are always numbers, i, v, relatively prime to 2D, of such nature that for all primes, p and q, which satisfy the congruence p =, q a= 7 (mod 8SD1), the Pell equation t2 - pqDu2 = 1 possesses a fundamental solution, t = T, u = U, for which neither T + 1 nor T - 1 is divisible by pq. P. TANNERY, "L'extraction des racines carres d'apres Nicolas Chuquet," Bibliotheca mathematica, vol. I (2), p. 17, Stockholm, 1887. R. MtLLER, "Uber rationale Dreiecke und ihren Zusammenhang mit der Pell'schen Gleichung," Archiv der Mathematik und Physik, vol. V (2), p. 111, Leipzig, 1887. R. MARCOLONGO, "Sull' analisi indeterminata di 2~ grado," Giornale di matematiche, vol. XXV, p. 161, vol. XXVI, p. 65, Naples, 1887, 1888. B. H. RAU, H. PLAMENEWSKY, H. L. ORCHARD, "Solve in positive integers the equation x2 - 19y2 = 81," Mathematical questions from the Educational Times, vol. XLVIII, p. 48, London, 1888. J. PEROTT, "Sur l'equation t2 - Du2 = - 1," Journal fur die reine und angewandte Mathematik, vol. CII, p. 185, Berlin, 1888. The work of Lagrange, Legendre and Dirichlet is THE PELL EQUATION 133 reviewed. Many theorems are given concerning the possibility of solving the equation t2 - Du2 = -1 in special cases. The two following are interesting examples. The necessary and sufficient condition for the solution of the equation t2 - 2qu2 = - 1 when q is of the form 16n+9, is that 2(-1)/4 = - 1 (mod q). If q = 16n + 1, then 2(q-')/4 _- 1 (mod q) is necessary but not sufficient. For the equation t2 - 2qu2 = - I to be possible where q is a prime of the form 8n + 1, it is necessary in the decomposition q = c2 + 2d2, that d be divisible by 8. This condition is sufficient when q is of the form 16n + 9, but not when q is of the form 16n + 1. A. MARTIN, "An error in Barlow's theory of numbers," Bulletin of the Philosophical Society of Washington, vol. XI, p. 592, Washington, 1888. x2- 5,658y2 = 1 should be x2 - 56,587y2 = 1. With this typographical correction the solution of Barlow is correct. F. TANO, "Sur quelques theoremes de Dirichlet," Journal fur die reine und angewandte Mathematik, vol. CV, p. 160, Berlin, 1889. The solutions of the equation x2 - Ay2 = - are sought when A has any number of prime factors of the form 4n + 1. A. HURWITZ, "Uber eine besondere Art der Kettenbruchentwicklung reeler Gr6ssen," Acta mathematica, vol. XII, p. 367, Stockholm, 1889. The author applies a peculiar kind of continued fractions to the solution of the equation x2 - Ay2 = - 1. G. FRATTINI, "Sulla risoluzione dell' equazione x2 - (a2 + 1)2 = -= N in numeri interi," Periodico di matematica, vol. VI, p. 85, Rome, 1891. 134 THE PELL EQUATION From the fundamental solution, x0, yo, of the equation x2- (a2 + l)y2 = -N, all the solutions, x, y, of both equations are given by the identity x + a 1 ( + y = ( a + 1)(a + 4a2 + 1) in which n runs through all odd values. G. FRATTINI, "Dell' analisi indeterminata di secundo grado," Periodico di matematica, vol. VI, p. 169, vol. VII, p. 7, 49, 88, 119, 172, Rome, 1891, 1892. C. A. ROBERTS, A. MARTIN, "A table of the square roots of the prime numbers of the form 4n + 1 less than 10,000 expanded as periodic continued fractions," Mathematical magazine, vol. II, p. 105, Washington, 1892. Comment is made upon the fact that the number of terms in the period of the continued fraction for 4N never exceeds 2a where a2 is the largest square less than N, provided N < 1,000; and it is asked whether this can be proved generally. It can not. See the numbers 8,269, 8,941, 9,949, 4,909. I have found several numbers smaller than these for which the period exceeds 2a, namely, 1,726, 1,831, 2,011. See the table in appendix A. G. FRATTINI, "Due propositioni della teoria dei numeri e loro interpretazione geometrica," Atti della Reale Accademia dei Lincei, vol. I (5), p. 51, Rome, 1892. The author gives several theorems for deducing integral solutions of the equations 2 - Dy2 = N, together with geometric interpretations. G. FRATTINI, "A complemento di alcuni teoremi del sig. Tchebicheff," Atti della Reale Accademia dei Lincei, vol. I (5), p. 85, Rome, 1892. This article discusses the solutions in positive integers of the equations x2 - Dy2 = - N. The values of x in THE PELL EQUATION 135 the successive solutions of the equation x2 - Dy2 = 1 are deduced from the series of numbers which separate the successive solutions of the given equation. E. LEMOINE, "Resolution complete des equations indeterminees, x2 + 1 = 2y2; x2- = 2y2," Coimbra, 1893; also Jornal de sciencias mathematicas e astronomicas, vol. XI, p. 68, Coimbra, 1892. K. SCHWERING, "Zerfallung der lemniskatischen Theilsungsgleichung in vier Factoren," Journal fur die reine und angewandte Mathematik, vol. CX, p. 42, vol. CXII, p. 37, Berlin, 1892, 1894. The solution of the Pell equation is connected with the theory of circle division and the theory of quadratic remainders. H. WEBER, "Ein Beitrag zur Transformationstheorie der elliptischen Functionen mit einer Anwendung auf Zahlentheorie," Mathematische Annalen, vol. XLIII, p. 185, Leipzig, 1893. The Pell equation is used to furnish the solutions of certain modular equations called Schlafi'sche equations, u = f(w) and v = f(nw), where n - 1 (mod 24) and is prime and greater than 3. As an example of the connection of the Pell equation with elliptic functions the equation x2 - 745y2 = 1 is discussed. A. CAYLEY, C. E. BICKMORE, A. R. FORSYTH, A. LODGE, J. J. SYLVESTER, "Tables connected with the Pellian equation," Report of the British Association, vol. LXII, p. 73, Nottingham, 1893, London, 1894, A. Cayley, "Papers," vol. XIII, p. 430, Cambridge, 1897. This table gives the fundamental solution of x2 - Ay2 = - 1 or of x2 - Ay2 = + 1 from A = 1,001 136 THE PELL EQUATION to A = 1,500. P. BACHMAN, "Zahlentheorie," vol. II, p. 92, Leipzig, 1894. G. DE LONGCHAMPS, "Sur certaines generalisations de l'equation de Pell," Journal de mathematiques elementaires, vol. XVIII, p. 5, Paris, 1894. The author discusses the equation Mx2 = Ay2 + Bz2 +... + Kt2 where M=A+B +C+.. +K admits an infinity of solutions, also the equations x2 - xy + y2 = 2 and 2 = y2 + p2. E. MAILLET, "Notes on a generalized Pell equation," Association frangaise pour l'avancement des sciences, comptes rendus des congres, p. 233, Paris, 1895. E. BORTOLOTTI, "Sulla frazioni continue algebriche periodiche," Rendiconti del Circolo Matematico di Palermo, vol. IX, p. 136, Palermo, 1895. The author discusses the periodicity which exists only under the necessary and sufficient condition that the Pell equation x2 - Ay2 = 1 in integral polynomials x, y, is solvable. G. SPECKMAN, " Fundamentalaufilsungen der Pell'schen Gleichung," Archiv der Mathematik und Physik, vol. XIII (2), p. 327, Leipzig, 1895. G. SPECKMAN, "Uber die Auflisung der Pell'schen Gleichung," Archiv der Mathematik und Physik, vol. XIII (2), p. 330, Leipzig, 1895. A great number of the solutions of the Pell equation THE PELL EQUATION 137 can be obtained from the following formulas, (a2 rn m)2 - (a2 2m)a2 = m2, (na2: m)2 - (n2a2 = 2nm)a2 = m2, (a, m, n = 1, 2, 3, *. )). H. W. L. TANNER, "Notes on the automorphs of binary quadratic forms," Messenger of mathematics, vol. XXIV, p. 180, London, 1895. Attention is drawn to the essential identity of automorphs with the theory of units in the generalized theory of numbers. The relation of the Pell equation to the automorph is thus more clearly shown. C. J. DE LA V. POUSSIN, "Sur les fractions continues et les forms quadratiques," Annales de la Societe Scientifiques de Bruxelles, vol. XIX, p. 111, Brussels, 1895. The author points out the advantage of the continued fractions of which all the partial quotients after the first are negative integers. G. FRATTINI, "Dell equazione di Pell a coefficiente algebrico," Giornale di matematiche di Battaglini, vol. XXXIII, p. 371, vol. XXXIV, p. 98, Naples, 1895, 1896. This is a treatise on the equation x2 - Ay2 = 1 in which A is an integral function of u, thus x2 - (au2 + bl + c)y2 = 1. G. SPECKMAN, "Uber unbestimmte Gleichungen xten Grades," Archiv der Mathematik und Physik, vol. XIV (2), p. 443, Leipzig, 1896. From the consideration of the solutions of the Pell equation of the second degree, T2 - DU2 = 1, formulas are produced for the solution of the equation TX -DUx = m2. A. PALMSTROM, "Quelques proprietes des solutions de certaines equations indeterminees de deuxieme degre," Aarbog, 1896. 138 THE PELL EQUATION The author discusses the law of recurrence which connects the different solutions of the Pell equation. C. ST6RMER, "Om en egenskab ved losningerne af den Pellske ligning x2 - Ay2 = - 1," Nyt Tidsskrift for Mathematik, vol. VII, p. 49, Copenhagen, 1896. The following theorem is found: if x and y are positive roots of the equation x2 - Ay2 = - 1, and a, b, the smallest among these roots, then 1 1 a tan-' - tan-' = 2 tan-' X2 n-1 X2n+l X2n and 1 1 b tan-1 + tan- = 2 tan-1-. X2n-1 X2 nl1 Y2 n C. STORMER, A. PALMSTROM, To solve in integers 1 + x2 = 2y4, L'Intermediare des mathematiciens, vol. III, p. 197, vol. IV, p. 89, Paris, 1896, 1897. Solutions are found by the aid of the Pell equation. A. BELIGNE, H. BROCARD, "Solution of X2 + (X + 1)2 = y4 L'Intermediare des mathematiciens, vol. IV, p. 214, Paris, 1897. R. FRICKE, FELIX KLEIN, "Vorlesungen fiber die Theorie der elliptischen Modulfunctionen," vol. I, p. 253, Leipzig, 1897. M. CURTZE, "Quadrat- und Kubikwurzeln bei den Greichen nach Herons neu aufgefunden MerTpLK&," Zeitschrift fir Mathematik und Physik, Historisch-literarische Abteilung, vol. XLII, p. 113, Leipzig, 1897. M. CURTZE, "Die Quadratwurzelformel des Heron bei Regimontan und damit Zusammenhangendes," Zeitschrift fur Mathematik und Physik, Historisch-literarische Abteilung, vol. XLII, p. 145, Leipzig, 1897. THE PELL EQUATION 139 A. PALMSTROM, "Generalized Pellian equations," L'Intermediare des mathematiciens, vol. IV, p. 169, Paris, 1897. The author says he has found properties of the equation, Xi X2 X3 * * * Xn-1 Xn XnA Xl X2 ' ' Xn-2 Xn-1 Xn-lA xnA xl... xn-3 Xn-2 xn-2A Xn-_A xnA... Xn-4 Xn-3 x2A x3A x4A... xnA xl for n > 2. For n = 2, it is the Pell equation. A. BOUTIN, "Developpement de /ix en fraction continue," Mathesis, vol. VII (2), p. 8, Paris, 1897. This article contains 35 formulas for the development of the square root of a number into a continued fraction. On this subject see also E. Lucas in Journal de mathematiques speciales, p. 1, Paris, 1887, and Journal fuir die reine und angewandte Mathematik, vol. XI, p. 332, Berlin, 1834, where there are 42 formulas of this kind. A. THUE, "Une solution de l'equation x2 - Dy2 = m," Archiv for Mathematik og Naturvidenskab, vol. XIX, p. 27, Christiania, 1897. C. STORMER, "Quelques theoremes sur l'equations de Pell x2 - Dy2 = 1, et leur applications," VidenskabsSelskabets Skrifter, No. 2, pp. 48, Christiania, 1897. One theorem deals with the integral solutions for y of the equations x2 - Dy2 = - 1, every prime divisor of which divides D. There are several theorems upon numbers of the form x2 + 1 with applications to the theory of the Pell equation. G. DE ROCQUIGNY, "Trouver deux entiers consecutis dont Fun soit un carre et l'autre un triangulaire," Mathesis, vol. VII (2), p. 279, Paris, 1897. 140 THE PELL EQUATION The above problem is solved by Emmerich and A. Goulard, Mathesis, vol. VIII (2), p. 52, Paris, 1898. H. Brocard adds a bibliographic note, op. cit., p. 112. Besides questions already referred to, see E. Catalan in Nouvelle correspondance mathematiques, p. 194, Brussels, 1877, and p. 285, 1879; G. de Longchamps, Journal de mathematiques elementaires, vol. VIII, p. 15, Paris, 1884; E. Catalan, H. Brocard, A. Boutin, Journal de mathematiques speciales, vol. XVII, p. 23, 117, 139, Paris, 1893. C. ST6RMER, Inquiry concerning tables of solutions of the Pell equations x2 - Dy2 = =t 1, L'Intermediare des mathematiciens, vol. IV, p. 123, Paris, 1897. Particular inquiry is made for the equation x2 _ Dy2 = - where D = 28p1p2... pnql2q22 * qn2 in which 5 = 0 or 1 and the p's and q's are of the form 4n + 1. E. B. ESCOTT, "Reply concerning tables of solutions of Pell equations," L'Intermediare des mathematiciens, vol. V, p. 276, Paris, 1898. The writer mentions no solutions but refers to tables of continued fractions in Martin's Mathematical magazine, vol. II, No. 7, 1892. See the reference. H. BROCARD, "Note bibliographique sur l'equation de Pell," Mathesis, vol. VIII (2), p. 112, Paris, 1898. A. BOUTIN, "Sur un equation de Pell," Mathesis, vol. VIII (2), p. 159, Paris, 1898. The equation discussed is x2- (m2- l)y2 = 1, particularly in the case m = 2. E. DE JONQUIERES, "Sur un point de doctrine dans la theorie des formes quadratiques," Comptes rendus de l'Academie, vol. CXXVI, p. 991, 1077, Paris, 1898. THE PELL EQUATION 141 The author compares his method for the study of quadratic forms with that of Gauss and Dirichlet and shows the superiority of his own method, for example in the solution of the equation (a2 - 4)y - 4x2 = 1. E. DE JONQUIERES, "Formules generales donnant des valeurs de D pur lesquelles l'6quation t2 - Du2 = - 1 est resoluble en nombres entiers," Comptes rendus de l'Academie, vol. CXXVI, p. 1837, Paris, 1898. Two of the theorems proved are: the equation is solvable when D is of the form 4n2 + n + 5; when D = a2(n2 + 1) and n is a multiple of a, the equation is impossible. G. WERTHEIM, "Uber die Ausziehung der Quadrat- und Kubikwurzeln bei Heron von Alexandria," Zeitschrift fiir mathematischen und naturwissenschaftlichen Unterricht, vol. XXX, p. 253, Leipzig, 1899. A. PALMSTROM, "Les solutions ordinaires des equations x2 + 1 = 2y2 forment elles le systeme complet des solutions?" L'Intermediare des mathematiciens, vol. VI, p. 40, Paris, 1899. Puisse, op cit., vol. V, p. 31, 1898, asks a question implied in the above. See J. A. Serret, "Cours d'algebre superieure," p. 77, 5th ed., Paris, 1885, and P. Bachman, "Die Elemente der Zahlentheorie," p. 182, Paris, 1892. A. PALMSTROM, "Proprietes relatives a l'6quation x2 - Ay2 = 1," L'Intermediare des mathematiciens, vol. VI, p. 210, Paris, 1899. The properties discussed are X2n + Y2n A = (X1 + Y1 IA)2n = (Xn + Yn A)2, X2n = Xn2 + AYn2 = 2Xn2 - 1, and Y2n = 2XYn. G. WERTHEIM, "Pierre Fermats Streit mit John Wallis, ein Beitrag zur Geschichte der Zahlentheorie," Abhand 142 THE PELL EQUATION lungen zu Geschichte der Mathematik, vol. IX, p. 555, Leipzig, 1899. E. CAHEN, "Resolution de l'equation de Pell" in "Elements de theorie des nombres," p. 221-275, Paris, 1900. The solutions are based on the theory of quadratic forms. Theorems are given for both positive and negative discriminant. From the two positive solutions all the others are deduced. A. CUNNINGHAM, "If r and z are two integral solutions of 2r2 - 2 = 1, then 13 + 33 + 53 + *. + (2r - l)3=r2z2, Mathematical questions from the Educational Times, vol. LXXII, p. 45, vol. LXXIII, p. 132, London, 1900. R. W. D. CHRISTIE, A. CUNNINGHAM, "If p is a prime of the form 8M + 3, solve X - pY2 = 1 in integers without using the method of continued fractions, generalize the method for all odd primes, state the cases when the solution is instanstaneous," Mathematical questions from the Educational Times, vol. LXXIII, p. 115, London, 1900. The equation X2 - pY2 = 1 may sometimes be solved by assuming X, Y, to be functions of other variables x, y, the transformed equation in x, y, admitting of easy solution. Many examples are given. C. DE POLIGNAC, "Solution of question No. 14,713," Mathematical questions from the Educational Times, vol. LXXV, p. 67, 1901. If ti, ul, is the fundamental solution of the Pell equation t2 - Du2 = 1, and t,, u,, is any other solution, there is a linear substitution (Qlx + S1) xl (Plx + R1)' such that when we write the nth power (Qnx + Sn) axr = (Pn + Rn) THE PELL EQUATION 143 then Qn and Pn/ul give the solution, tn, u,, of the above equation. H. KONEN, "Geschichte der Gleichung t2 - Du2 = 1," Leipzig, 1901. This work is well conceived and developed with talent. It gives an especially fine account of the contest between Fermat and the English mathematicians. A third of the entire treatise is given over to the discussion of the work of Lagrange. In the bibliography of twenty-four references given at the close the following corrections should be noted: A. Martin's article is in the Analyst, vol. IV, p. 154 (1877), not in vol. V, p. 118; and the article in vol. V, p. 118 (1878), is by D. S. Hart, "The solution of an indeterminate problem." Meyer's article should be dated 1887 not 1889, and Schmidt's 1874 not 1876. Speckman's two articles in the Archiv der Mathematik und Physik are on p. 327 and p. 330, 1895, not on p. 216 and p. 130, 1894. Frattini's (not Trattini) first. article is in volume XXXIII of the Giornale di Matematiche, not vol. XXX. The list contains the following references which I have not found elsewhere: Berkhan, "Lehrbuch der unbestimmten Analytik," 2 Bd., Halle, 1855-1856, Bd. II, p. 121. E. Meissel, "Beitrag zur Pellschen Gleichung h6herer Grade," Progr. der Ober-Realschule zu Kiel, 1891. Pistor, "Uber die Auflisung der unbestimmten Gleichung 2. Grades in ganzen Zahlen," Programm, Hamm, 1833. G. W. Tenner, "Einige Bemerkungen fiber die Gleichung ax2 == 1 = y2," Programm, Merseburg, 1841. G. RICALDE in L'Intermediare des mathematiciens, vol. VIII, p. 256, Paris, 1901. The identities (k2n 1)2- n(k2n - 2)k2 = 1, n2 - (n2- 1) 12 = 1, (2n2 + 1)2 - (n2 + 1)(2n)2 = 1, (8n + 25)2 - (4n2 + 25n + 39).42 = 1, 144 THE PELL EQUATION {8[n3 + (n + 1)3] + 1}2 - [(2n + 1)2 + 4]{4[n3 + (n + 1)3][n2 + (n + 1)2]}2 = 1, give solutions besides 1, 0; for the equation x2 - Ay2 = 1, where A has one of the forms n2 1, n(k2n 2), 4n2 + 25n + 39, (2n + 1)2 + 4. The author asks whether there is an analogous identity for every value of A not a perfect square. H. BROCARD, On a law of recurrence characteristic of all the solutions of the Pell equation, L'Intermediare des mathematiciens, vol. VIII, p. 59, Paris, 1901. G. RICALDE, E. B. ESCOTT, "La loi de recurrence des solutions de l'equation de Pell," L'Intermediaire des mathematiciens, vol. VIII, p. 286, Paris, 1901. A. BOUTIN, "Resolution complete de l'equation x2- (Am2 + Bm + C)y2= 1 ouf A, B, C, sont des entiers, par une infinite des polyn6mes en m," L'Intermediare des mathematiciens, vol. IX, p. 60, Paris, 1902. The equations x2 - Ay2 = - 1, when we have A a conveniently chosen function of the second degree of the parameter m, are solved completely by an infinity of polynomials in m. These polynomials satisfy certain differential equations of the second order. For example, if we have the equations 2 - (m2 + )y2 = 1, x2 (m2 + )y2 = 1, the recurring series Yo =, y1 = 1,. * y, = 2myn-i + yn-2, Xo = 1, X1 = m,..* * n = 2mxn-l + Xn-2 (n = 2,3,...), with x, y, of even index give solutions of the first equation and x, y, of odd index to the second equation; and the THE PELL EQUATION 145 functions of m, Xn, y, satisfy the differential equations (M2 + 1)dmY + 3mdn (n2- l)y = 0, (m2 + 1) dm2 + m - n2Xn = 0. As another example, the equations x2- (25m2 - 14m + 2)y2 = 1, x2- (25m2 - 14m + 2)y2 = - 1, with m a positive integer, have for solutions members of the following series with even and odd index respectively, yo = 0, Y1 = 5,. yn = 2(25m - 7)yn-1 + yn-2, x0 = 1, x = 25m - 7, * X.n = 2(25m- 7)x,_1 + xn-2, and we have ~d2yn dyn (25m2-14m+2) dn +3(25m-7) d - 25(n2 - l)y 0 = 0, d2x dx (25m2-14mq-2) d - + (25m - 7) dmn 25n2x = 0. J. ROMERO, "Equations de Pell," L'Intermediare des mathematiciens, vol. IX, p. 182, Paris, 1902. If x, y, is a solution of one of the equations x2- Ay2 = = 1, we have identically (ny2 = x)2 - (n2y2 - 2nx + A)y2 = -- 1. A. WEREBRUSOW, "Equations de Pell," L'Intermediare des mathematiciens, vol. IX, p. 182, Paris, 1902. In the equations x2 - Ay2 = - 1, A may have the form a2m2 + 2bm t- c, provided the determinant b2 - a2C = t 1. ll~~~b 146 THE PELL EQUATION P. TANNERY, Review of "Geschichte der Gleichung t2 - Du2 = 1," by H. Konen, Bulletin des sciences mathematique, vol. XXVII (2), p. 47, Paris, 1903. A HOLM, "On the convergents to a recurring continued fraction with application to find integral solutions of the equation x2 - Cy2 = (- 1)nDD," Proceedings of the Edinburgh Mathematical Society, vol. XXI, p. 163, London, 1903. The solutions of the above equation are furnished by, (1), when c the number of quotients in the cycle of i/C is even, x - y XC = (pn - qn ]C)(pc -q sC) where pn, qn, is a solution of the equation x2 - Cy2 = Dn. n any integer, pc, qC, is the fundamental solution of the equation x2 - Cy2 = 1, m any integer or 0; (2) when c is odd, x - Y XC = (pn, - qn C) (p - qc WC)2n, m any integer or 0. RUDIS, WEREBRUSOW, BESOUCLEIN, "Solution de lequation x2 - Dy2 = - 1," L'Intermediare des math6 -maticiens, vol. X, p. 102, p. 224, Paris, 1903. G. DE LONGCHAMPS, "Equation x2 - Dy2 = - 1," L'Intermediare des mathematiciens, vol. X, p. 319, Paris, 1903. E. B. ESCOTT, A. WEREBRUSOW, "The necessary relations between the integers, a, b, c, * * in order that the expression [a, b, c,.. c, b, a] [b, c,...c, b] where fN = (a, b, c,... c, b, 2a,...), be an integer," L'Intermediare des mathematiciens, vol. X, p. 97, 318, vol. XI, p. 154, Paris, 1903, 1904. THE PELL EQUATION 147 The conditions are deduced from the theory of the Pell equations x2 - Ny2 = - 1. H. BROCARD, P. F. TEILHET, "The equation X2 - ay2 =- _2," L'Intermediare des mathematiciens, vol. X, p. 277, vol. XII, p. 81, Paris, 1903, 1905. R. W. D. CHRISTIE, A. CUNNINGHAM, R. F. DAVIs, A. H. BELL, "Without any continued fraction procedure to solve x2 - 149y2 = 1 and to generalize the method," Mathematical questions from the Educational Times, vol. VI (2), p. 87, London, 1904. [Nearly the same in the Educational Times, vol. LVI, p. 269, London, 1903, and vol. LVII, p. 200, 275, London, 1904.] Some of these methods remind us of Brahmagupta. Conformal division is used. See A. Cunningham, "On the connection of quadratic forms," Proceedings of the London Mathematical Society, vol. XXVIII, p. 289, London, 1896. Bell makes use of a combination of triple series of arithmetical progressions. Christie uses primitive roots. R. W. D. CHRISTIE, "Primitive roots applied to the Pellian equation," Mathematical questions from the Educational Times, vol. VI (2), p. 98, London, 1904. R. W. D. CHRISTIE, "In the Pellian equation with any prime of the form 4m + 1 and convergents, p,', qn, as usual, prove qn2 + qn+2 = q2n++," Educational Times, vol. LVII, p. 41, London, 1904. See under Euler on p. 68. Example, X2 - 277y2 = - 1, then (22,213)2 + (6,524)2 = 535,979,945, where (8,920,484,118)2- 277(535,979,945)2 = - 1. 148 THE PELL EQUATION R. W. D. CHRISTIE, E. B. ESCOTT, "Prove that the socalled Pellian equation may be reversed by the dual principle as follows: Since (4a3 + 6a2 + 6a + 2)2 - (4a2 + 4a + 5)(2a2 + 2a + 1)2=- 1 for all values of the letter, then (2a3 + 6a2 + 6a + 4)2 - (5a2 + 4a + 4)(a2 + 2a + 2)2= -a6. Give a few other examples and show how to introduce the prime roots," Mathematical questions from the Educational Times, vol. VI (2), p. 119, London, 1904. A general solution is {k(4n2a2:= 4na + 4n2 + 1) + (2na =F a + 2n)}2 + 1 = {(2na =F 1)2 + (2n)2} {(2kn + 1)2 + (2kna + k -+ a)2}2, where all the letters are arbitrary. P. F. TEILHET, A. BOUTIN, "The form 632 + 1 is not a square if 13 is a root of 72 - 332 = 1," L'Intermediare des mathematiciens, vol. XI, p. 68, 182, Paris, 1904. E. B. ESCOTT, "Solutions de l'equation 2 - Ay2 =- a," L'Intermediare des mathematiciens, vol. XI, p. 156, Paris, 1904. A condition necessary that the equation (ax - by)2 - Dy2 = a may be solvable in integers is ( a' a where a', a", a"',... are odd prime factors of a; but this condition is not sufficient. G. FRATTINI, "Applicazione di un concetto nuovo all' analisi indeterminata aritmetica e algebrica di 2~ grado THE PELL EQUATION 149 con una nota sull' equazione di Pell," Periodico di matematica, vol. I (3), p. 57, Leghorn, 1904. In introducing the concept index, meaning the maximum number of factors of an irrational binomial, the author solves by a new method the equation x2 - Dy2 = N in the case where D is an integer or a polynomial of even degree. A. S. WEREBRUSOW, "Solution de x2 - Ay2 = - 1," L'Intermediare des mathematiciens, vol. XI, p. 154, 242. Paris, 1904. G. FRATTINI, "Nota sull' equazione di Pell," Periodico di matematica, vol. XIX, p. 71, Leghorn, 1904. R. W. D. CHRISTIE, A. CUNNINGHAM, "To find an infinity of positive integral values of X which belong to the same Y in the Pellian equation Xn2 - PY = - 1," Mathematical questions from the Educational Times, vol. VII (2), p. 79, London, 1905. Example, 702 - 293 132 = 74 - 1,. R. W. D. CHRISTIE, A. H. BELL, A. CUNNINGHAM, "Solve X2 - 19Y2 = - 3 in integers by the use of other convergents than the ordinary Pellian," Mathematical questions from the Educational Times, vol. VIII (2), p. 28, London, 1905. Use is made of Euler's theorem extended, of the negative development of 1/19, and of the triple series of Bell. A. CUNNINGHAM, "(1) Factorize into prime factors N = (70,600,7342 + 1). Here N = qp2 where p is a large prime, (2) Show how to find very large numbers (> 1050) of the form N = y2 + 1 = qm2 where m is very large (> 1025)," Mathematical questions from the Educational Times, vol. VIII (2), p. 83, London, 1905. 150 THE PELL EQUATION Published solutions of the Pellian equation y2 - Dx2 = _ 1 give this. See reference to British Association Report for 1893. From one such solution of the equation y2 - Dx2 = -1, we get (y2 + Dx2)2 - D(2xy)2 and multiplying these by conformal multiplication, y2 - DX2 = - 1, and here N1 = y12 + 1 = Dx12, yi = Y(Y2 + 3Dx2), xl = x(3y2 + Dx2) and the new N1, y1, X1, are much larger than before. By repeated conformal multiplication such numbers can be raised indefinitely. K. KOMMERELL, "Die ganzzahligen positiven L6sungen der unbestimmten Gleichung xyz(x+y-z)=t2," Mathematisch-naturwissenschaftliche Mitteilungen, vol. VII (2), p. 74, Stuttgart, 1905. The values of x, y, z, t, can be set forth in the following manner: T - a(d2yl — e2z1) x 2 = ad2yi, z = ae2z, t = adey1zlU2, if a, d, e, yi, zi, are any numbers chosen at pleasure (only d2yi, and e2z1 are relatively prime; yj and zi without square factors), and T, U, are any solution of the equation T2 - DU2 = 02 where D = 4yiz1, o = a(d2y, - e2z). A. AUBRY, "Theorie de l'equation de Pell," Mathesis, vol. V (3), [vol. XXV], p. 233, Paris, 1905. Several theorems are given in regard to the solvability THE PELL EQUATION 151 of the equations x2 - Ay2 = - 1, with applications. E. B. ESCOTT, "Solution de l'6quation x2 - Dy2 = - 1," L'Intermediare des mathematiciens, vol. XII, p. 53, Paris, 1905. If D = a2 + b2, it is a necessary but not a sufficient condition for the solution of the equation x2 - Dy2 = - 1 that a or b be a quadratic residue of D. The equation x2- 2,306y2 = - 1 has no solution although both 41 and 25 are quadratic residues of 2,306. RUDIS, "Solution de l'6quation x2- Dy2 =- 1, L'Intermediare des mathematiciens, vol. XII, p. 54, Paris, 1905. J. SANDIER, "Solution de l'equation x2 - Dy2 = - 1," L'Intermediare des mathematiciens, vol. XII, p. 249, Paris, 1905. This article contains seven theorems concerning the solution of the equations x2 - Dy2 = = 1. To obtain one solution of the equation x2 - Dy2 = N it is sufficient to have one each of the equations x2 - Dy2 = pi, (i = 1, 2, 3,...), p; being the distinct prime factors of N. J. SCHRiDER, "Eine Eigentiimlichkeit der Naherungswerte von /22," Archiv der Mathematik und Physik, vol. IX (3), p. 206, Leipzig, 1905. P. EPSTEIN, "Zu der Mitteilung von Herrn J. Schroder iber die Naherungswerte von J2," Archiv der Mathematik und Physik, vol. IX (3), p. 310, Leipzig, 1905. The results given are known from the theory of the Pell equation. If the continued fraction K=1+ —1 k +k+" k +$ 152 THE PELL EQUATION has the convergents Pa/Qa (a = 1, 2, 3,.*), the relation (1 - 1)a = (- )la-(kQ- Pa) exists only for k = 2. G. CANDIDO, "Sulle equazioni x2 - ay2 = zx, x2 - ay2 = b5," Giornale di matematiche di Battaglini, vol. XLIII, vol. XII (2), p. 93, Naples, 1905. This alludes to question 461 in L'Intermediare des mathematiciens, vol. II, p. 308, Paris, 1895. M. ELPHINSTONE, "History of India," p. 142, Note 16, 9th ed., E. B. Crowell, editor, London, 1905. The editor compares the work of Diophantus with that of Brahmagupta and the first solution of Lord Brouncker with that of Bhaskara. R. W. D. CHRISTIE, "Solve generally, in integers, the Diophantine equation y2 = av2 + avy + 1, where a is arbitrary," Mathematical questions from the Educational Times, vol. X (2), p. 24, London, 1906. There are solutions by the proposer, by A. Cunningham and by A. Holm, but all involve the Pell equation. A. HOLM, "If x = p, y = q, is any particular solution of x2 - Cy2 = 4 D, where C is a positive integer not a perfect square and D any integer, then all the [positive] integral solutions are furnished by x - y ~C = (p - q C)(r - s C)n, where n is any integer positive or negative and x = r, y = s, any particular solution of the equation 2 - Cy2 1," Mathematical questions from the Educational Times, vol. X (2), p. 29, London, 1906. This theorem will often enable one to dispense with Lagrange's chain of reductions for the case when D > ~/C. The solution is by B. Krishnamachari. THE PELL EQUATION 153 R. W. D. CHRISTIE, "Prove that the Pellian equation n2 - PYn2 = 1 can be put into a trigonometrical form, and give the general solution, thus for n = 3, X3 = cos 30 = 4 cos3 0 - 3 cos 0 = 4x' - 3x, Y3 = sin 30 = 4(cos2 0 - 1) sin 0 = 4x2y - y," Mathematical questions from the Educational Times, vol. IX (2), p. 11l, London, 1906. Let Xn 2 py2 = 1 then 2n - 2n-3n 2 n-n(n - 3) -4 Y^ __Xn __ n-2 n-4 -0! 1! 2! 2-7n(n - 4)(n - 5) 6 3! x ~ '. 2n-2m-ln(n - m)! nm -~ mm!(n-2m)! 2n-1 2n-3(n - 2) y xn-ly 1! -3y 2n- (n - 3)(n - 4) --! xn-a-y - * *. 2! - 2n-2m-l(n -m- 1)!xn2m 0 m!(n - 2m - 1)! which coincides with X = cos nO, Y = sin nO, when cos nO is developed from cosn 0, and sin nO is developed in the form sin 0 cosn 0 = sin 0 [(2 cos 0) n1 - (n - 2(2 cos 0) n(n - 3(n 4) (2 cos 0) ~.]. + 2n3( (2 COS )n-5 A. H. BELL, J. BLAIKIE, F. N. MAYERS, "When is a triangular number a pentagonal number?" Mathematical questions from the Educational Times, vol. IX (2), p. 40, London, 1906. This depends on the solution of the equation x2 - 3y2 = 1. 154 THE PELL EQUATION R. W. D. CHRISTIE, "Where p, q, are any convergents of Pn - 2q2 = 1, prove 2 tan-1 qn tan-1 _ tan-l = qn+l P2 n-1 4 P n 1 1 2 tan-1 pi = tan-' = tan-'1 = - pnd-l p2n-l 4 and show that Euler's and Machin's formulae are simple cases," Mathematical questions from the Educational Times, vol. IX (2), p. 52, London, 1906. A. CUNNINGHAM, A. HOLM, "Solve in integers x2 + y2 = 5z2: and generally solve x2 + y2 = Az2 where A is not a square number," Mathematical questions from the Educational Times, vol. IX (2), p. 69, London, 1906. The possibility of the solution of x2 + y2 = Az2 requires that t2 - A 2 = - 1 should be possible, and the actual solution, x, y, z, requires that t, u, be known. Holm says that A must either be a square or the sum of two squares. E. MALO) H. BROCARD, "Equations indeterminee, x2 - (y2_ 1-)p2 = 1, L'Intermediare des mathematiciens, vol. XIII, p. 228, Paris, 1906. If x = y, p = 1; if x = 2y2- 1, p = 2y. Other solutions are deduced from the ordinary formulas. RUDIS, "Solution de l'equation x Dy2 = 1," L'Intermediare des mathematiciens, vol. XIII, p. 243, Paris, 1906. Rudis has found that for D = a2 + b2 up to 394 there are 30 values for which x2 - Dy2 = - 1 is solvable. I have found that of the 110 composite numbers D = a2 b THE PELL EQUATION 155 between 1,501 and 2,000 there are 38 values for which the equation x2 - Dy2 = 1 is solvable. E. MALO, "Solution de l'6quation x2- Dy2 = - 1, L'Intermediare des mathematiciens, vol. XIII, p. 246, Paris, 1906. Several cases are discussed. The equation is impossible if D = 4u2 2 2, possible if D = 2[u2 + (u2 + 1)] or (2u + 1)2 + 4. M. VON THIELMANN, "Die Zerlegung von Zahlen mit Hilfe periodischer Kettenbriiche," Mathematische Annalen, vol. LXII, p. 401, Leipzig, 1906. Upon the ground of the period of the continued fraction of ik a Pellian equation is formed whose solution makes it possible to find the so-called type of the given number k, according to which the separation follows easily when possible. A. CUNNINGHAM, "High Pellian factorisations," Messenger of mathematics, vol. XXXV (2), p. 166, London, 1906. Under the designation "Pellian factorisations" is understood such separation of numbers N of the form N = x2 + 1 into factors as are easily worked out from the known solutions of the Pell equation. The existing tables of the solutions of the Pell equation give at first glance the separation of a considerable number of very large numbers N. These numbers are called Pellian numbers. This article shows the factorization of N up to 78 figures. R. W. D. CHRISTIE, A. H. BELL, "Having given n2 - pyn2= 1 to find a multiplier giving successive values, 156 THE PELL EQUATION one having been found. What is the simplest multiplier for x2 - 601y2 = 1?" Mathematical questions from the Educational Times, vol. XI (2), p. 39, 54, London, 1907, Educational Times, vol. LIX, p. 350, Aug. 1, 1906, p. 412, Sept. 1, 1906, London. If one solution, x, y, is known for the equation 2 - = x2 py2 1, then since (2x2 _ 1)2 2)2 = 4x2(x2 _ py2) _ 4x2 + 1 = 1 we may use 2x as a multiplier. In general xn+1 = 2x, - x_-1, Y n+-l = 2XnYU - yn-1_ For 601 the multiplier is 2 (38,902,815,462,492,318,420,311,478,049). R. W. D. CHRISTIE, "If 2 - py2 = 1 and X2 - pY2= 1, prove (X)2 + Y2 = (xy)2 + y2 for an infinity of integral values of x, y, X, Y," Mathematical questions from the Educational Times, vol. XI (2), p. 96, London, 1907. This is proved by A. Cunningham as well as the proposer. x2 -1 X2 - 1 y2 =- P = y (Xz - 1)2 = (X2 - )y2. A. CUNNINGHAM, "Let En = 22'", Fn = E + 1, Gn = E- 1, show that every Fn ( > 5) may be expressed algebraically in the form 2Fn = t2 - GnU2 and obtain the expression. Discuss the possibility of obtaining a second expression, 2F,, = t'2 - GnU2, which together with the former shall be suitable for factorizing F,." Mathematical questions from the Educational Times, vol. XII (2), p. 21, London, 1907. THE PELL EQUATION 157 The Pell equation is here made serviceable in factorization. A. CUNNINGHAM, "On the factorization of the Pellian terms," British Association Report, p. 462, London, 1907. J. SOMMER, "Vorlesung iiber Zahlentheorie," Leipzig, 1907. In obtaining the units of a quadratic domain, f( m.), where the fundamental number, m, is real, the solution of the Pel] equation is necessary. Proof is given that in the case of a real domain the fundamental unit can always be found and from it the other units may be obtained. This corresponds exactly to the proof of the solvability of the equations x2 - my = 1 and x2 +x+ [ ( — ) ]2=1 (p. 102). This work has been revised and translated into French by A. Levy, "Introduction a la theorie des nombres algebriques," Paris, 1911. B. NIEWENGLOWSKI, "Note sur les equations x2 - ay2 = 1 et x2 -ay2 = -1," Bulletin de la Societe Mathematique de France, vol. XXXV, p. 126, Paris, 1907. The equation x2- ay2 = 1 represents in rectangular axes an hyperbola. Integral points are defined as points of this hyperbola having integral numbers for co6rdinates. If M and M1 are two integral points, the parallel to the tangent at M1 drawn through M meets the hyperbola again in an integral point. Other similar properties are proved for both equations. P. A. MACMAHON, "The Diophantine equation xn - Nyn = z," Proceedings of the London Mathematical Society, vol. V (2), p. 45, London, 1907. 158 THE PELL EQUATION For the cases n = 2, 3, 4, this equation leads to problems of the invariant theory. G. CORNACCHIA, "Sulla risoluzione in numeri interi dell' equazione x" -qyn = 1," Revista di fisica, matematica e scienze naturali, vol. VII (2), p. 221, Pavia, 1907. If the equation possesses an integral solution x, y, then x/y is a convergent of 1q developed into a continued fraction. AURIC, "Sur le developpement en fraction continue d'une irrationnelle ambigue du second degre," Bulletin de la Societe Mathematique de France, vol. XXXV, p. 121, Paris, 1907. The discussion depends upon the Pell equation t - Du2 = 4 R. W. D. CHRISTIE, A. H. BELL, "Find, by the quintic roots of minus unity, all the values in succession of x, y, in the equation x2 - 5y2 = - 4," Mathematical questions from the Educational Times, vol. XIII (2), p. 35, London, 1908, Educational Times, vol. LX, p. 353, Aug. 1, 1907, London. x = (w2 + CO3)2n + (C4 + c5)2n = 3, 7, 18, *., (CO2 + 23)n2 - (W4 + WC5 )2n Y (C(02 + C03) - (C4 +- ) - 1 ' R. W. D. CHRISTIE, T. STUART, A. CUNNINGHAM, "Let x2 - py2 = 1, where p is of the form 4m + 3, then (x\ (py - y + z z )-P (P+ 1) ) z being (2py - p + 1)/(p + 1)," Mathematical questions from the Educational Times, vol. XIV (2), p. 56, London, 1908. THE PELL EQUATION 159 From one integral solution a double infinity of fractional solutions are obtained. The prime or composite nature of p has no effect upon the theorem. E. B. ESCOTT, A. CUNNINGHAM, "The solutions of Tn2 - 2Un2 = (- 1)n form two recurring series, Tn = 1, 1, 3, 7, 17,..., Un = 0, 1, 2, 5, 12, *., whose scale of relation is Un+2 = 2un+1 + u,, factor completely U84, a number of thirty-two digits," Mathematical questions from the Educational Times, vol. XIV (2), p. 105, London, 1908. A. BOUTIN, "Developpement de JIN en fraction continue et resolution des equations de Fermat," Association frangaise pour l'avancement des sciences, congres de Clermont-Ferrand, p. 18, Paris, 1908. K. H. VAHLEN, E. CAHEN, "Resolution de l'equation de Fermat," Encyclopedie des sciences mathematiques, Tome I, vol. 3, p. 107, Paris, 1908. See also Encyklopadie der mathematischen Wissenschaften, vol. I, part 2, p. 599, Leipzig, 1904. DUBouIS, E. B. ESCOTT, Divisors of the form x2 - ay2, L'Intermediare des mathematiciens, vol. XV, p. 172, vol. XVI, p. 81, Paris, 1908, 1909. A. CUNNINGHAM, "Let t'2 - 2u2 = - 1, t2 - 22 = 1. Resolve u66 into its prime factors," Mathematical questions from the Educational Times, vol. XV (2), p. 95, London, 1909. This number contains fifty-one figures. G. FONTENE', "Sur un cas particulier de l'equation de Pell," Bulletin de mathematiques elementaires, vol. XIV, p. 209, Paris, 1909. The author shows how if the smallest solution of the equation x2 - Ay2 = 1 is known, all the others may be determined. 160 THE PELL EQUATION A. CHATELET, "Sur un cas particulier de l'6quation de Pell," Bulletin de mathematiques elementaires, vol. XIV, p. 307, Paris, 1909. The author shows that it is possible always to obtain one solution of x2 - Ay2 = 1. R. W. D. CHRISTIE, JAGAT CHANDRA PAL, A. CUNNINGHAM, "Prove the following equations, (PnPn+1)2 +- (2qnQn+i)2 = q2n+1 pnn+1 - 2ql7q = + 1, where pn2 - 2q,2 = 1, hence show that an endless number of equations can be formed such as 32 42 = 52, 202 212 = 292, * * Mathematical questions from the Educational Times, vol. XV (2), p. 74, London, 1909. G. FRATTINI, "La nozione d'indice e l'analisi indeterminata dei polinomi interi," Atti del IV congresso internazionale dei matematici, vol. II, p. 178, Rome, 1909. If D and N are integral polynomials in a, then if one solution, a, a, of the Pell equation x2 - Dy2 = 1 is known, all the solutions of the equation x2 - Dy2 = N can be deduced, and by this means we obtain the notion of an index of a binomial E + F AiD where E and F are rational functions of a. A. AUBRY, "L'oeuvre arithmetique d'Euler," L'Enseignement mathematique, vol. XI, p. 329, Paris, 1909. G. FONTENE, "Sur un cas particulier de l'6quation de Pell," Bulletin de mathematiques elementaires, vol. XV, p. 65, Paris, 1910. A. LEVY, "Sur un cas particulier de l'equation de Pell," and "Sur une application de l'equation de Pell," Bulletin de mathematiques elementaires, vol. XV, p. 66, Paris, 1910. THE PELL EQUATION 161 The author makes application to the equation X2 + y2 = z2 when two of the three unknown are consecutive integers. E. B. ESCOTT, T. HAYASHI, "The equation 2 - Dy2 = 4zn." L'Intermediare des mathematiciens, vol. XV, p. 153, vol. XVII, p. 2, 137, 229, 253, Paris, 1908, 1910. R. BRICARD, "Le produit de huit nombres entiers consecutifs ne peut etre un carre parfait," L'Intermediare des mathematiciens, vol. XVII, p. 139, Paris, 1910. The theory of the Pell equation is used in the demonstration that the product of eight consecutive integers can not be a perfect square. L. W. REID, "The elements of the theory of algebraic numbers," p. 423, New York, 1910. Pell's equation is discussed in connection with the determination of the units of a real quadratic "realm." D. N. RANUCCI, "Risoluzione dell' equazione xn - Ayn = = 1 con una nuova dimostrazione dell' teorema di Fermat," Rome, 1911. 12 162 THE PELL EQUATION APPENDIX A. The following table gives the development of 4A into a continued fraction for non-square values of A from A = 1,501 to A = 2,012. The development is periodic, the period beginning with the second term. The last term of the period is always double the first term of the development. The period is symmetric. The next to the last term equals the first, the third from the last equals the second, and so on. Then only the first half of the period need be written. To indicate its position in the period, the term that completes the first half is written in parentheses. When however the number of terms is odd, the middle term and the one preceding it are enclosed in parentheses. Under each term a number is written which is the absolute value of the form x2 - Ay2 when x/y is the convergent corresponding to that point in the development of the continued fraction. These values are alternately positive and negative. The 1 which would always come under the first term of the continued fraction is omitted. Example, 41,512 = 38 + -1,512 - 38, 1 1,512 + 38 41,512 - 30 1,512 - 38 68 1 68 68 _1,512 + 30 41,512 - 33 41,512- 30 9 9 9 _ 11,512 + 33 41,512 - 14 41,512 - 33 47 47 47 _ 1,512 + 14 41,512 - 14 1,512 -14 28 = 1+ 28 l1,512- 14 28 28 THE PELL EQUATION 163 28 1,512 + 14 '1,512 - 33 /1,512- 14 47 47 Therefore ~~1 +~- — 1 1,~512 = 38 + ---- 1+- — 1 1 + 1_1 7+ 1+ 71+ —1 1 + 7+ 76 + 1+76+..*, or it might be written, 41,512 = (38; 1, 7, 1, (1), 1, 7, 1, 76;..), 1, 68, 9, 47, (28), 47, 9, 68, 1, *..; and in the table it is written, 38 1 7 1 (1) 68 9 47 (28) The convergents are 1 38 39 311 350 661 1,011 7,738 8,749 O' 1' 1' 8' 9' 17' 26' 199' 225' and 12- 1,512.02 = + 1, 382 - 1,512 12 - 68, 392 - 1,512.12 + 9, 3112 - 1,51282 = -47, 8,7492 - 1,512-2252 = + 1, 1501 38 1 2 1 7 1 6 6 3 4 1 5 1 1 1 4 1 1 14 57 20 53 9 60 11 12 23 15 55 12 45 29 44 15 36 41 5 1 18 2 3 2 1 1 2 1 1 25 (4) 69 4 33 20 27 36 37 25 33 44 3 (19) 164 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1522 THE PELL EQUATION 38 1 3 10 1 4 1 1 1 2 (38) 58 19 7 59 14 43 31 38 29 (2) 38 1 3 3 (8) 59 18 23 (9) 38 1 3 1 1 2 1 4 (2) 60 17 39 36 23 48 15 (32) 38 1 3 1 6 3 1 (14) 61 16 55 11 19 56 (5) 38 1 4 5 2 1 10 2 2 (38) 62 15 14 25 50 730 31 (2) 38 1 4 1 1 3 1 3 (3) 63 14 39 37 18 49 19 (22) 38 1 (4) 64 (13) 38 1 5 2 18 1 (24) 65 12 35 4 71 (3) 38 1 6 12 1 4 3 1 7 1 (6) 66 11 6 59 15 19 54 9 61 (10) 38 1 6 1 3 1 2 3 5 1 2 6 1 2 1 1 14 1 (37) 67 10 55 17 46 25 22 13 47 26 11 50 23 34 43 5 71 (2) 38 1 7 1 (1) 68 9 47 (28) 38 1 8 1 2 1 48 2 3 (4) 69 8 53 21 49 16 9 32 21 (17) 38 1 10 7 1 2 (4) (4) 70 7 10 49 25 (17) (17) 38 1 (11) 71 (6) 38 1 14 1 1 2 2 1 2 1 1 5 1 10 3 1 1 1 1 72 5 44 33 27 25 43 24 35 41 12 61 7 21 40 33 35 36 1 9 8 1 1 4 1 1 125 341631 31 45 8 9 40 37 15 45 28 49 3 24 15 56 11 20 47 2 1 (18) 21 55 (4) 38 1 18 (2) 73 4 (37) 38 1 (24) 74 (3) 38 1 (37) 75 (2) 38 (1) (76) 39 (78) (1) THE PELL EQUATION 165 1523 39 (39) (2) 1524 39 (26) (3) 1525 39 19 (1) (1) 4 (39) (39) 1526 39 15 1 1 1 1 2 1 1 (10) 5 46 31 35 38 25 34 43 (7) 1527 39 (13) (6) 1528 3911 6 2 2 1 3 1 7 1 (8) 7 12 31 24 47 17 56 9 63 (8) 1529 39 9 1 3 4 1 1 1 2 2 15 4 1 1 (6) 8 55 19 16 43 31 40 25 32 5 17 37 40 (11) 1530 39 8 1 2 (8) 9 49 26 (9) 1531 39 7 1 4 2 1 12 2 1 4 1 1 5 2 8 4 4 2 1 10 57 15 25 51 6 27 46 15 38 39 13 34 9 18 17 26 45 3 2 2 1 25 2 1 1 1 15 (39) 19 30 23 54 3 29 39 30 47 5 (2) 1532 39 7 9 1 1 1 3 1 (18) 11 8 47 29 44 17 59 (4) 1533 39 6 1 1 19 (26) 12 37 41 4 (3) 1534 39 (6) (13) 1535 39 5 1 1 2 2 (7) 14 41 35 26 31 (10) 1536 39 5 4 1 2 3 (19) 15 16 47 25 23 (4) 1537 39 4 1 7 1 10 3 5 1 2 (2) 16 57 9 64 7 24 13 49 24 (29) 1538 39 4 1 1 1 1 (38) 17 41 34 31 47 (2) 1539 39 4 2 1 8 (39) 18 25 50 9 (2) 1540 39 4 8 (2) 19 9 (35) 1541 39 3 1 10 2 6 1 1 1 15 19 1 1 3 2 2 2 1 (2) 20 55 7 35 11 47 28 49 5 4 43 35 20 29 28 25 44 (23) 1542 39 3 1 2 1 1 1 (38) 21 46 23 42 29 49 (2) 166 THE PELL EQUATION 1543 39 3 1 1 3 1 3 1 5 3 1 25 2 2 1 12 2 1 1 22 37 39 18 51 17 54 13 19 58 3 33 23 53 6 29 38 33 1 2 2 1 1 8 7 (39) 39 26 27 34 43 9 11 (2) 1544 39 3 2 2 (9) 23 28 31 (8) 1545 39 3 3 1 4 6 1 (14) 24 19 51 16 11 64 (5) 1546 39 3 7 (1) (1) 25 10 (39) (39) 1547 39 (3) (26) 1548 39 2 1 9 5 1 (18) 27 49 8 13 63 (4) 1549 39 2 1 3 1 25 2 4 1 3 8 2 15 3 1 2 6 5 11 19 28 45 17 60 3 35 15 51 20 9 36 5 21 44 27 12 15 7 4 1 1 2 3 2 1 5 1 6 3 3 1 1 1 (1) (1) 45 33 28 21 25 49 12 59 11 23 20 41 33 36 (35) (35) 1550 39 2 1 2 2 1 3 (2) 29 41 26 25 46 19 (31) 1551 39 2 1 1 1 1 2 1 1 (6) 30 37 35 34 39 25 35 42 (11) 1552 39 2 1 1 8 6 2 (4) 31 33 44 9 12 33 (16) 1553 39 2 2 4 1 1 9 (3) (3) 32 29 16 37 41 8 (23) (23) 1554 39 2 2 1 1 1 (10) 33 25 41 30 47 (7) 1555 39 2 3 3 1 6 2 2 12 1 2 1 4 1 7 1 (14) 34 21 19 54 11 30 31 6 55 21 51 14 59 9 66 (5) 1556 39 2 4 6 1 (18) 35 17 11 65 (4) 1557 39 2 5 1 1 2 1 8 19 1 1 1 1 2 (8) 36 13 41 36 23 52 9 4 47 31 36 37 29 (9) 1558 39 2 8 3 1 1 1 3 1 2 1 (38) 37 9 21 42 31 43 18 49 21 57 (2) 1559 39 2 15 3 2 1 2 2 5 1 1 1 7 4 (39) 38 5 23 26 43 25 31 13 46 29 47 10 19 (2) 1560 39 (2) (39) 1561 39 1 125 1 5 8 1 1 1 1 2 1 2 4 1 9 15 1 2 40 39 3 64 13 9 45 32 35 39 24 42 27 15 59 8 5 53 24 2 1 4 1 1 3 4 1 1 1 (10) 25 48 15 40 37 21 16 45 29 48 (7) THE PELL EQUATION 167 1562 39 1 1 10 1 3 1 (2) 41 38 7 58 17 49 (22) 1563 39 1 1 6 1 2 5 1 2 1 3 (39) 42 37 11 49 26 13 51 22 47 21 (2) 1564 39 1 1 4 1 3 2 1 9 5 5 1 7 1 (18) 43 36 15 52 19 25 51 8 15 13 60 9 67 (4) 1565 39 1 1 3 1 1 1 19 (7) (7) 44 35 19 44 29 49 4 (11) (11) 1566 39 1 1 2 1 15 8 1 2 1 2 (2) 45 34 23 54 5 9 53 22 45 25 (29) 1567 39 1 1 2 2 3 (39) 46 33 27 29 23 (2) 1568 39 1 1 2 (19) 47 32 31 (4) 1569 39 1 1 1 1 3 5 1 4 2 3 1 2 1 1 9 3 15 1 1 48 31 35 40 21 13 36 15 32 19 47 24 35 43 8 25 5 40 39 10 1 4 (26) 7 59 16 (3) 1570 39 1 1 1 1 1 8 5 (1) (1) 49 30 39 31 46 9 14 (39) (39) 1571 39 1 1 1 2 1 15 7 1 6 3 (39) 50 29 43 22 55 5 10 61 11 25 (2) 1572 39 1 1 1 5 2 3 3 (6) 51 28 47 13 32 21 23 (12) 1573 39 1 1 1 19 (6) 52 27 51 4 (13) 1574 39 1 2 15 1 1 6 1 2 3 3 1 7 5 1 (38) 53 26 5 41 38 11 50 25 22 19 55 10 13 65 (2) 1575 39 1 2 (5) 54 25 (14) 1576 39 1 2 3 8 11 10 1 4 2 1 1 1 2 1 1 4 1 2 55 24 23 9 39 40 7 60 15 28 39 33 40 25 36 41 15 49 24 2 (19) 33 (4) 1577 39 1 2 2 6 1 3 1 4 5 19 11 (4) 56 23 32 11 56 17 53 16 13 61 8 7 (19) 1578 39 1 2 1 1 1 1 1 (12) 57 22 41 33 38 31 47 (6) 1579 39 1 2 1 3 1 12 2 5 5 8 1 1 1 3 7 1 2 15 1 58 21 50 17 59 6 35 14 15 9 47 30 43 21 10 49 27 5 42 1 4 1 3 1 1 2 10 1 25 1 1 2 1 2 (39) 37 15 53 18 41 35 30 7 70 3 45 34 25 42 29 (2) 1580 39 1 (2) 59 (20) 168 THE PELL EQUATION 1581 39 1 3 5 19 1 2 (4) 60 19 15 4 53 25 (17) 1582 39 1 3 2 3 5 1 (4) 61 18 31 22 13 57 (14) 1583 39 1 3 1 2 3 1 4 1 10 1 1 5 1 1 2 (39) 62 17 47 26 19 53 14 61 7 41 38 13 43 34 31 (2) 1584 39 1 (3) 63 (16) 1585 39 1 4 3 8 1 1 6 1 2 2 4 (1) (1) 64 15 24 9 40 39 11 51 24 31 16 (39) (39) 1586 39 1 4 1 (2) (2) 65 14 49 (25) (25) 1587 39 1 5 7 (13) 66 13 11 (6) 1588 39 1 5 1 1 1 8 4 1 6 2 3 12 12 4 1 67 12 49 29 48 9 16 57 11 33 19 48 23 44 27 16 43 1 1 1 25 1 (18) 37 31 48 3 73 (4) 1589 39 1 6 3 1 5 2 1 2 15 1 1 2 1 19 4 1 68 11 20 53 13 28 41 29 5 44 35 23 55 4 17 44 3 3 2 1 (10) 20 25 52 (7) 1590 39 1 (6) 69 (10) 1591 39 1 7 1 7 11 3 1 2 2 3 2 1 1 1 25 (1) 70 9 65 10 7 21 46 25 30 21 27 41 30 49 3 (74) 1592 39 1 (8) 71 (8) 1593 39 1 10 2 (2) 72 7 32 (27) 1594 39 1 12 3 8 1 1 4 1 3 1 7 (5) (5) 73 6 25 9 41 38 15 54 17 57 10 (15) (15) 1595 39 1 (14) 74 (5) 1 1; 33 36 1 1 1 43 1596 39 1 (18) 75 (4) 1597 39 1 25 1 1 1 8 4 1 1 2 2 2 6 4 19 1 2 1 5 76 3 51 28 49 9 17 41 36 27 28 31 12 19 4 57 21 53 12 1 10 1 1 (3) (3) 63 7 43 36 (21) (21) 1598 39 1 (38) 77 (2) 1599 39 (1) (78) THE PELL EQUATION 169 1601 40 (80) (1) 1602 40 (40) (2) 1603 40 26 1 2 8 1 1 3 1 2 (5) 3 53 26 9 42 37 19 46 27 (14) 1604 40 (20) (4) 1605 40 (16) (5) 1606 40 13 2 1 8 4 2 1 26 (40) 6 27 50 9 18 25 54 3 (2) 1607 40 11 2 3 1 2 1 6 1 1 4 5 1 1 (39) 7 34 19 49 22 53 11 41 38 17 14 37 43 (2) 1608 40 (10) (8) 1609 40 8 1 9 7 5 4 1 4 1 1 5 1 1 1 1 1 15 2 2 9 65 8 11 15 16 55 15 39 40 13 45 32 39 31 48 5 33 25 1 2 1 1 1 2 2 1 (26) (26) 45 24 41 33 40 27 24 55 (3) (3) 1610 40 (8) (10) 1611 40 7 3 1 1 (39) 11 22 35 45 (2) 1612 40 6 1 2 8 1 1 2 1 26 (20) 12 49 27 9 43 36 23 56 3 (4) 1613 40 (6) (6) (13) (13) 1614 40 5 1 2 1 1 1 15 2 3 2 1 (12) 14 51 23 43 30 49 5 34 21 25 53 (6) 1615 40 5 2 1 8 (4) 15 26 51 9 (19) 1616 40 (5) (16) 1617 40 4 1 2 1 1 4 (2) 17 49 24 37 41 16 (33) 1618 40 4 2 5 (3) (3) 18 33 14 (23) (23) 1619 40 4 4 2 15 1 1 1 5 11 3 7 1 2 1 1 1 1 1 19 17 35 5 50 29 47 14 7 25 10 53 23 41 34 37 35 38 1 1 (39) 31 49 (2) 1620 40 (4) (20) 170 THE PELL EQUATION 1621 40 3 1 4 1 1 1 1 1 1 2 3 1 5 1 15 3 1 26 11 21 52 15 44 33 37 36 35 39 28 19 55 12 65 5 20 59 3 7 2 6 1 5 3 19 1 4 2 2 1 1 8 2 1 3 6 2 36 11 60 13 25 4 63 15 31 27 35 44 9 28 45 21 12 33 3 (1) (1) 20 (39) (39) 1622 40 3 1 1 1 5 1 1 3 1 2 3 7 (40) 22 43 31 46 13 41 38 19 47 26 23 11 (2) 1623 40 3 2 (26) 23 34 (3) 1624 40 3 2 1 8 3 1 (10) 24 25 52 9 20 57 (7) 1625 40 3 4 1 2 (2) (2) 25 16 49 25 (29) (29) 1626 40 3 11 5 3 2 7 1 1 1 2 1 1 2 1 (12) 26 7 15 22 33 10 47 31 42 25 38 39 23 55 (6) 1627 40 2 1 (39) 27 53 (2) 1628 40 2 1 (6) 28 49 (11) 1629 40 2 1 3 2 1 2 1 1 6 1 3 5 1 19 2 1 15 2 29 45 20 27 44 25 36 43 11 55 20 13 65 4 27 52 5 37 (8) (9) 1630 40 2 1 2 81 1 2 13 (16) 30 41 29 9 45 34 31 6 (5) 1631 40 2 1 1 2 (5) 31 37 38 29 (14) 1632 40 2 1 1 (19) 32 33 47 (4) 1633 40 2 2 3 2 4 37 26 1 4 11 2 19 2 2 1 8 3 33 29 21 32 17 24 11 3 63 16 7 27 51 8 33 24 53 9 21 1 (2) 48 (23) 1634 40 2 2 1 2 1 4 (40) 34 25 46 23 50 17 (2) 1635 40 2 3 2 1 (4) 35 21 26 49 (15) 1636 40 2 4 3 1 4 1 1 1 2 2 1 6 26 1 4 2 3 15 36 17 20 53 15 45 32 41 27 25 51 12 3 64 15 32 23 5 1 8 (20) 68 9 (4) 1637 40 2 5 1 2 1 2 19 1 6 (2) (2) 37 13 52 23 44 29 4 67 11 (31) (31) THE PELL EQUATION 171 1638 40 2 (8) 38 (9) 1639 40 2 15 1 2 (3) 39 5 54 25 (22) 1640 40 (2) (40) 1641 40 1 1 (26) 41 40 (3) 1642 40 1 1 11 13 2 2 1 1 1 2 1 8 3 (1) (1) 42 39 7 6 33 26 41 33 42 23 54 9 22 (39) (39) 1643 40 1 1 6 (1) 43 38 11 (62) 1644 40 1 1 4 1 9 3 7 (20) 44 37 15 61 8 25 11 (4) 1645 40 1 1 3 1 3 3 1 1 2 19 1 8 (16) 45 36 19 51 20 21 41 36 31 4 69 9 (5) 1646 40 1 1 3 (40) 46 35 23 (2) 1647 40 1 1 (2) 47 34 (27) 1648 40 1 1 2 8 1 1 1 1 1 4 6 1 1 (4) 48 33 31 9 47 32 39 33 44 17 12 41 39 (16) 1649 40 1 1 1 1 4 2 9 1 (2) (2) 49 32 35 43 16 35 8 53 (25) (25) 1650 40 1 1 1 1 1 (2) 50 31 39 34 41 (25) 1651 40 1 1 1 2 1 1 2 2 3 8 1 2 14 26 1 7 (6) 51 30 43 25 39 38 27 30 23 9 55 22 51 17 3 69 10 (13) 1652 40 1 1 1 4 2 (2) 52 29 47 16 31 (28) 1653 40 1 1 10 1 19 2 2 (2) 53 28 51 7 71 4 33 28 (29) 1654 40 1 2 (40) 54 27 (2) 1655 40 1 2 7 (16) 55 26 11 (5) 1656 40 1 2 3 1 (2) 56 25 20 49 (23) 1657 40 1 2 2 2 8 1 1 1 2 1 2 1 4 2 1 4 9 1 26 57 24 29 32 9 48 31 43 24 47 23 51 16 27 48 17 8 71 3 (4) (4) (19) (19) 1658 40 1 2 1 1 (4) (4) 58 23 38 41 (17) (17) 172 THE PELL EQUATION 1659 40 1 2 1 2 1 1 (26) 59 22 47 25 35 46 (3) 1660 40 1 2 1 8 3 3 1 1 3 1 2 1 1 1 1 2 5 (20) 60 21 56 9 24 21 39 40 19 49 24 41 35 36 39 29 15 (4) 1661 40 1 3 11 2 1 1 5 1 2 15 1 19 2 3 1 1 2 2 61 20 731 35 44 13 49 28 5 73 4 35 20 41 37 28 25 1 (6) 52 (11) 1662 40 1 3 3 3 2 1 1 (26) 62 19 23 22 29 34 47 (3) 1663 40 1 3 1 1 5 3 1 2 2 1 1 1 12 1 26 3 1 5 63 18 39 41 14 21 47 26 27 42 31 49 6 73 3 21 54 13 1 1 11 8 1 (39) 39 42 7 9 71 (2) 1664 40 1 3 1 4 3 2 1 (19) 64 17 55 16 23 25 55 (4) 1665 40 1 4 8 1 6 1 1 (8) 65 16 9 64 11 40 41 (9) 1666 40 1 4 (2) 66 15 (34) 1667 40 1 4 1 5 2 4 2 1 10 1 (39) 67 14 59 13 34 17 26 53 7 73 (2) 1668 40 1 5 3 2 1 1 1 1 (26) 68 13 23'28'39 37 32 49 (3) 1669 40 1 5 1 4 1 1 2 3 1 2 3 1 15 1 1 3 26 1 69 12 59 15 43 36 29 20 47 27 19 60 5 45 36 23 3 75 19 2 6 3 8 1 3 (5) (5) 4 37 12 25 9 57 20 (15) (15) 1670 40 1 6 2 3 1 5 (16) 70 11 34 19 55 14 (5) 1671 40 1 7 5 3 (13) 71 10 15 25 (6) 1672 40 1 8 (10) 72 9 (8) 1673 40 1 9 4 4 1 6 1 1 1 2 5 1 (10) 73 8 19 16 59 11 47 32 41 29 13 64 (7) 1674 40 1 10 1 2 2 1 4 8 1 7 3 (2) 74 7 54 25 26 49 17 9 65 10 23 (31) 1675 40 1 12 1 1 1 8 2 3 2 2 1 5 7 3 (1) 75 6 51 29 50 9 34 21 31 25 51 14 11 21 (50) 1676 40 1 15 2 1 1 2 1 2 9 1 6 1 1 5 1 3 4 (20) 76 5 31 37 40 25 44 29 8 65 11 41 40 13 55 20 19 (4) 1677 40 1 19 (2) 77 4 (39) THE PELL EQUATION 173 1678 40 1 26 3 8 1 3 2 2 1 1 2 4 6 13 2 (39) 78 3 26 9 58 19 31 27 39 38 29 18 13 6 39 (2) 1679 40 1 (39) 79 (2) 1680 40 (1) (80) 1682 41 (82) (1) 1683 41 (41) (2) 1684 41 27 2 1 8 2 4 2 1 4 2 3 1 1 1 6 1 4 1 1 3 28 51 9 35 17 27 49 16 33 20 45 31 48 11 60 15 44 35 1 1 15 1 4 5 3 1 2 1 1 11 6 1 3 (20) 33 48 5 63 16 15 21 48 25 36 45 7 12 55 21 (4) 1685 41 20 (1) (1) 4 (41) (41) 1686 41 16 2 2 2 1 7 1 1 (40) 5 33 29 25 53 10 39 43 (2) 1687 41 13 1 2 8 1 3 1 2 27 (41) 6 53 27 9 59 1847 29 3 (2) 1688 41 11 1 2 1 1 1 (9) 7 56 23 44 31 49 (8) 1689 41 10 3 1 4 2 1 1 1 1 1 15 1 4 1 1 5 1 3 8 21 53 16 29 40 35 39 32 49 5 64 15 40 41 13 56 19 2 11 3 2 1 (26) 35 7 24 25 56 (3) 1690 41 9 (8) 9 (10) 1691 41 8 4 1 2 2 16 (41) 10 17 50 25 34 5 (2) 1692 41 7 (2) 11 (36) 1693 41 6 1 5 2 8 1 2 6 1 1 20 27 2 1 1 (1) (1) 12 61 13 36 9 52 27 12 39 43 4 3 31 39 36 (37) (37) 1694 41 6 3 7 1 (10) 13 25 10 67 (7) 1695 41 5 1 6 1 (1) 14 61 11 49 (30) 1696 41 5 2 11 3 4 1 4 1 2 8 1 3 1 (19) 15 36 7 25 16 57 15 49 28 9 60 17 63 (4) 1697 41 5 7 3 2 3 1 9 (1) (1) 16 11 23 32 19 59 8 (41) (41) 1698 41 4 1 5 11 1 1 1 1 (40) 17 57 14 7 47 34 33 49 (2) 174 THE PELL EQUATION 1699 41 4 1 1 3 5 4 1 1 1 15 1 5 2 2 27 13 1 2 18 41 39 22 15 17 47 30 51 5 66 13 31 33 3 6 55 25 2 1 2 2 7 1 4 1 1 1 1 2 7 9(41) 27 45 26 33 10 61 15 45 34 37 39 30 11 9 (2) 1700 41 4 3 (20) 19 25 (4) 1701 41 4 8 1 (10) 20 9 68 (7) 1702 41 3 1 11 27 2 2 1 1 3 2 1 8 (2) 21 58 7 3 34 27 38 41 21 26 53 9 (37) 1703 41 3 1 2 1 5 (6) 22 49 23 53 14 (13) 1704 41 3 1 1 2 1 (2) 23 40 39 25 47 (24) 1705 41 3 (2) 24 (31) 1706 41 3 3 2 2 1 2 1 7 (1) (1) 25 22 31 26 47 23 55 10 (41) (41) 1707 41 3 6 (41) 26 13 (2) 1708 41 3 (20) 27 (4) 1709 41 2 1 15 1 6 1 1 2 1 3 2 2 1 1 20 11 1 3 28 53 5 68 11 44 37 25 49 20 31 28 35 47 4 7 59 20 4 1 (1) (1) 17 44 (35) (35) 1710 41 2 1 5 (4) 29 49 14 (19) 1711 41 2 1 2 1 13 16 2 8 1 2 2 (2) 30 45 23 57 6 5 38 9 54 25 30 (29) 1712 41 2 1 1 1 11 (5) 31 41 32 49 7 (16) 1713 41 2 1 1 2 1 5 1 1 1 4 1 1 9 1 3 1 (26) 32 37 41 24 53 13 48 31 47 16 39 43 8 61 17 64 (3) 1714 41 (2) (2) (33) (33) 1715 41 2 2 2 1 3 1 1 1 (7) 34 29 26 49 19 46 31 49 (10) 1716 41 2 2 1 4 2 (6) 35 25 51 16 35 (12) 1717 41 2 3 2 (4) 36 21 33 (17) 1718 41 2 4 2 1 1 1 2 1 (40) 37 17 29 41 34 43 23 59 (2) THE PELL EQUATION 175 1719 41 2 5 1 7 2 4 7 3 5 1 1 1 1 (8) 38 13 63 10 35 18 11 25 14 45 35 34 47 (9) 1720 41 2 8 1 2 1 1 (3) 39 9 55 24 39 41 (20) 1721 41 2 16 10 3 4 1 1 3 1 4 2 2 (7) (7) 40 5 8 25 17 41 40 19 55 16 31 32 (11) (11) 1722 41 (2) (41) 1723 41 1 1 27 5 1 8 2 1 1 3 2 1 3 1 11 13 1 3 42 41 3 14 63 9 31 37 42 21 27 49 18 61 7 6 59 21 (41) (2) 1724 41 11 11 2 1 3 29 1 15 1 2 2 1 1 1 2 (20) 43 40 7 29 47 20 35 8 71 5 56 25 28 41 35 40 31 (4) 1725 41 1 1 7 20 1 1 1 (2) 44 39 11 4 51 31 44 (25) 1726 41 1 1 5 27 1 1 16 9 5 1 4 1 2 2 1 2 1 1 1 45 38 15 3 42 41 59 14 59 15 51 26 27 46 25 42 35 39 1 1 4 3 1 2 1 5 1 1 1 11 4 1 1 7 1 34 45 17 21 50 23 54 13 49 30 51 7 18 39 43 10 57 3 13 1 1 2 3 1 (40) 21 6 47 35 30 19 63 (2) 1727 41 1 1 3 1 (6) 46 37 19 58 (11) 1728 41 1 1 3 8 1 (19) 47 36 23 9 71 (4) 1729 41 1 1 2 1 1 2 1 2 1 8 1 1 27 5 (6) 48 35 27 39 40 25 48 23 56 9 40 43 3 16 (13) 1730 41 1 1 2 5 (1) (1) 49 34 31 14 (41) (41) 1731 41 1 1 1 1 6 1 (26) 50 33 35 46 11 70 (3) 1732 41 1 1 1 1 1 1 2 2 7 6 1 4 27 1 1 5 1 8 2 51 32 39 37 36 41 27 33 11-12 59 17 3 44 39 13 64 9 32 2 (20) 33 (4) 1733 41 1 1 1 2 2 4 11 1 1 1 20 6 2 1 (4) (4) 52 31 43 28 23 19 7 52 29 53 4 13 28 49 (17) (17) 1734 41 1 1 1 3 1 2 1 1 4 1 (40) 53 30 47 19 50 25 38 43 15 67 (2) 1735 41 1 1 1 (7) 54 29 51 (10) 1736 41 1 (1) 55 (28) 176 THE PELL EQUATION 1737 41 1 2 10 11 1 4 3 2 2 1 1 1 (8) 56 27 8 7 63 16 23 31 27 43 32 49 (9) 1738 41 1 2 4 1 1 3 13 1 1 1 1 2 11 1 1 8 1 (2) 57 26 17 42 39 23 6 49 33 38 39 31 7 42 41 9 57 (22) 1739 41 1 2 2 1 6 1 7 (2) 58 25 26 53 11 65 10 (37) 1740 41 1 2 2 (20) 59 24 35 (4) 1741 41 1 2 1 1 1 3 1 1 6 2 1 1 5 1 4 1 2 1 1 60 23 44 33 45 20 39 43 12 31 36 45 13 60 15 52 25 36 47 27 4 7 2 1 20 5 1 1 16 6 1 8 2 (2) (2) 3 20 11 27 55 4 15 39 44 5 12 65 9 33 (29) (29) 1742 41 1 2 1 4 (6) 61 22 53 17 (13) 1743 41 1 (2) 62 (21) 1744 41 1 3 5 3 6 1 1 1 4 1 11 9 (5) 63 20 15 25 12 49 31 48 15 64 7 9 (16) 1745 41 1 3 (2) (2) 64 19 (31) (31) 1746 41 1 3 1 1 1 (8) 65 18 47 31 50 (9) 1747 41 1 3 1 13 7 1 1 8 1 3 11 1 2 66 17 63 6 11 41 42 9 58 21 7 54 27 1 3 1 3 27 1 1 2 (41) 21 51 22 3 49 34 33 (2) 1748 41 1 4 (4) 67 16 (19) 1749 41 1 4 1 1 2 2 1 20 4 1 (6) 68 15 44 37 29 25 57 4 17 60 (11) 1750 41 1 (4) 69 (14) 5 1 1 1 2 4 4 47 33 42 29 18 1751 41 1 5 2 (4) 70 13 35 (17) 1752 41 1 (5) 71 (12) 1753 41 1 6 1 1 1 1 127 3 2 4 1 4 9 10 2 1 3 1 72 11 48 33 41 32 51 3 24 33 16 57 17 9 8 29 48 19 51 2 1 (2) (2) 24 47 (27) (27) 1754 41 1 7 2 1 1 2 1 3 11 1 2 (3) (3) 73 10 31 38 41 25 49 22 7 55 26 (23) (23) 1755 41 1 8 (3) 74 9 (26) THE PELL EQUATION 177 1756 41 1 9 2 (20) 75 8 39 (4) 1757 41 1 (10) 76 (7) 1758 41 1 (12) 77 (6) 1759 41 1 15 1 3 1 2 1 1 3 1 5 4 1 3 5 3 27 1 1 78 5 63 18 51 25 39 42 19 57 14 17 54 21 15 26 3 53 30 1 4 1 13 6 2 1 1 1 2 2 11 1 1 3 2 8 1 49 15 65 6 13 30 41 35 42 27 34 7 45 38 21 35 9 67 7 2 (41) 10 39 (2) 1760 41 1 (19) 79 (4) 1761 41 1 (26) 80 (3) 1762 41 1 (40) 81 (2) 1763 41 (1) (82) 1765 42 (84) (1) 1766 42 (42) (2) 1767 42 (28) (3) 1768 42 (21) (4) 1769 42 16 1 4 3 (6) (6) 5 65 16 25 (13) (13) 1770 42 (14) (6) 1771 42 (12) (7) 1772 42 10 1 1 (20) 8 41 43 (4) 1773 42 9 2 1 (8) 9 28 53 (9) 1774 42 8 2 2 2 1 27 2 1 2 7 3 1 1 8 1 3 1 3 1 10 33 30 25 58 3 31 43 30 11 23 38 45 9 61 18 55 19 46 1 1 3 5 2 1 13 2 1 4 1 16 (42) 33 45 22 15 27 55 6 29 50 15 66 5 (2) 1775 42 7 1 1 1 5 2 1 (2) 11 50 31 49 14 29 46 (25) 13 178 THE PELL EQUATION 1776 42 (7) (12) 1777 42 6 2 8 1 9 1 1 1 4 3 3 2 1 4 1 1 2 1 27 13 37 9 69 8 51 31 48 17 24 23 27 51 16 43 39 24 59 3 2 (1) (1) 32 (39) (39) 1778 42 (6) (14) 1779 42 5 1 1 1 1 2 1 3 3 2 1 1 (41) 15 46 35 37 42 25 50 21 23 30 35 49 (2) 1780 42 5 3 1 4 1 1 (20) 16 21 55 16 39 45 (4) 1781 42 4 1 20 (3) (3) 17 65 4 (25) (25) 1782 42 4'1 29 (42) 18 49 29 9 (2) 1783 42 4 2 3 4 2 2 27 1 2 1 6 113 4 1 8 1 1 2 19 33 23 18 31 34 3 61 22 57 11 69 6 17 62 9 46 37 27 1 1 1 1 (41) 41 38 33 51 (2) 1784 42 4 4 1 2 1 1 3 (10) 20 17 51 25 40 41 23 (8) 1785 42 (4) (21) 1786 42 3 1 4 1 7 1 1 1 2 13 1 2 2 4 1 1 5 11 22 55 15 63 10 49 33 42 31 6 57 25 33 17 41 42 15 7 1 8 (2) 70 9 (38) 1787 42 3 1 1 1 (41) 23 46 31 53 (2) 1788 42 3 1 1 (20) 24 37 47 (4) 1789 42 3 2 1 2 4 1 1 1 1 6 2 3 1 3 3 1 27 2 25 28 45 29 17 45 36 35 47 12 35 20 53 21 20 63 3 36 3 5 2 1 4 1 20 3 11 1 3 9 6 1 (16) (16) 23 15 28 51 15 67 4 27 7 60 21 9 12 69 (5) (5) 1790 42 3 4 (8) 26 19 (10) 1791 42 3 8 7 1 1 2 1 5 1 (3) 27 10 11 45 38 25 54 13 59 (18) 1792 42 (3) (28) 1793 42 2 1 9 1 11 5 4 1 3 1 1 1 5 1 (6) 29 53 8 71 7 16 17 56 19 47 32 49 13 64 (11) THE PELL EQUATION 179 1794 42 2 1 4 3 5 1 (2) 30 49 17 25 14 55 (23) 1795 42 2 1 2 1 1 2 4 13 1 8 2 (16) 31 45 26 41 39 30 19 6 71 9 39 (5) 1796 42 2 1 1 1 3 16 1 2 11 1 3 3 7 2 1 1 (20) 32 41 35 44 23 5 55 28 7 61 20 25 11 32 35 49 (4) 1797 42 2 1 1 3 1 6 3 1 1 5 1 20 2 1 7 (28) 33 37 44 19 59 12 23 39 44 13 69 429 52 11 (3) 1798 42 2 2 13 1 2 1 3 9 6 2 (2) 34 33 6 59 23 51 22 9 13 33 (29) 1799 42 2 2 2 2 1 (41) 35 29 31 25 59 (2) 1800 42 2 2 1 (8) 36 25 56 (9) 1801 42 2 3 1 1 5 10 2 3 16 1 2 4 1 27 2 11 1 1 37 21 40 43 15 8 35 24 5 56 27 16 67 3 40 7 51 32 1 2 2 1 4 3 1 4 1 8 1 (1) (1) 45 25 49 24 53 17 21 56 15 64 9 48 (35) (35) 1802 42 2 (4) 38 (17) 1803 42 2 6 (28) 39 13 (3) 1804 42 2 8 1 16 10 1 1 3 1 2 1 1 1 1 1 1 1 (20) 40 9 72 5 8 45 39 20 51 25 43 36 39 37 40 33 51 (4) 1805 42 2 (16) 41 (5) 1806 42 (2) (42) 1807 42 1 1 27 1 5 9 3 1 1 2 2 1 3 (6) 43 42 3 69 14 9 23 42 39 29 27 49 22 (13) 1808 42 1 1 11 1 1 1 (4) 44 41 7 52 31 49 (16) 1809 42 1 1 7 4 2 1 9 1 16 9 2 1 1 4 1 2 1 1 (2) 45 40 11 19 27 55 8 73 5 9 32 37 45 16 53 25 41 40 (27) 1810 42 1 1 5 5 1 8 1 1 (1) (1) 46 39 15 14 65 9 49 34 (39) (39) 1811 42 1 1 3 1 (41) 47 38 19 65 (2) 1812 41 1 1 3 5 (28) 48 37 23 16 (3) 1813 42 1 1 2 1 1 1 6 (2) 49 36 27 44 33 49 12 (37) 1814 42 1 1 2 4 11 1 16 8 2 5 1 1 1 1 2 3 (42) 50 35 31 19 7 74 5 10 37 14 47 35 38 41 29 25 (2) 180 THE PELL EQUATION 1815 42 1 1 1 1 (13) 51 34 35 49 (6) 1816 42 1 1 1 1 2 6 1 2 1 1 5 9 3 2 3 1 4 1 (9) 52 33 39 40 31 12 55 25 39 44 15 9 24 33 20 57 15 65 (8) 1817 42 1 1 1 2 11 1 4 10 2 4 1 5 1 (2) 53 32 43 31 7 64 17 8 37 16 61 13 56 (23) 1818 42 1 1 1 3 4 1 2 1 (8) 54 31 47 22 17 54 23 58 (9) 1819 42 1 1 1 5 18 1 1 1 2 5 3 4 2 2 1 4 1 (41) 55 30 51 13 66 9 50 33 43 30 15 25 18 33 26 53 15 69 (2) 1820 42 1 1 1 (20) 56 29 55 (4) 1821 42 1 2 16 1 2 1 3 3 6 1 4 6 2 1 3 1 1 2 57 28 5 60 23 52 21 25 12 61 17 13 29 49 20 43 39 28 2 20 1 11 4 5 2 4 (28) 35 4 75 7 20 15 35 19 (4) 1822 42 1 2 5 1 3 4 2 13 1 3 1 1 3 1 1 27 1 8 1 58 27 14 57 21 18 37 6 63 19 42 41 21 38 47 3 74 9 42 1 11 2 (42) 43 7 29 (2) 1823 42 1 2 3 2 1 1 1 4 2 1 1 5 1 (41) 59 26 23 29 43 34 47 17 31 37 46 13 71 (2) 1824 42 1 2 (2) 60 25 (32) 1825 42 1 2 (1) (1) 61 24 (41) (41) 1826 42 1 2 1 2 1 2 (42) 62 23 50 25 46 31 (2) 1827 42 1 2 1 (8) 63 22 59 (9) 1828 42 1 3 11 1 27 1 1 2 2 3 1 1 1 8 1 6 4 2 1 64 21 7 76 3 49 36 29 32 21 47 32 51 9 67 12 19 28 51 4 2 1 (20) 17 27 57 (4) 1829 42 1 3 3 2 7 2 1 11 1 1 6 16 1 20 2 3 1 3 65 20 23 35 11 28 55 7 44 41 13 5 77 4 37 20 55 19 1 2 1 1 1 2 (2) 52 25 44 35 43 28 (31) 1830 42 1 3 1 1 (16) 66 19 39 46 (5) 1831 42 1 3 1 3 3 1 1 1 2 8 5 1 1 2 2 2 5 1 67 18 55 21 22 45 35 42 31 10 15 45 38 29 30 33 14 53 2 3 13 1 27 1 1 2 11 1 4 1 3 1 2 16 1 27 25 6 77 3 50 35 33 7 66 15 58 19 49 30 5 62 THE PELL EQUATION 181 3 7 1 1 8 1 (41) 21 11 42 43 9 75 (2) 1832 42 1 4 (21) 68 17 (4) 1833 42 1 4 2 1 3 (28) 69 16 29 48 23 (3) 1834 42 1 4 1 2 1 1 2 5 3 9 4 1 13 2 8 (12) 70 15 54 25 42 39 31 15 26 9 17 65 6 39 10 (7) 1835 42 1 5 7 1 1 1 1 1 3 1 (7) 71 14 11 49 34 41 35 46 19 61 (10) 1836 42 1 5 1 1 1 110 9 2 2 1 (20) 72 13 47 36 35 49 89 35 25 59 (4) 1837 42 1 6 6 2 4 1 1 2 1 1 1 11 27 1 20 2 6 73 12 13 36 17 44 39 27 43 36 41 33 52 3 79 4 39 72 1 1 18 1 (6) 51 31 52 9 68 (11) 1838 42 1 6 1 4 5 1 11 2 2 3 2 (42) 74 11 62 17 14 67 7 34 31 22 37 (2) 1839 42 1 7 1 1 2 3 (28) 75 10 47 37 30 25 (3) 1840 42 1 8 1 1 5 (5) 76 9 44 41 15 (16) 1841 42 1 9 1 2 1 4 1 1 1 1 1 2 16 1 3 1 1 2 1 77 8 59 23 55 16 47 35 40 37 41 32 5 64 19 43 10 25 56 7 (12) 11 (7) 1842 42 1 11 3 1 1 1 5 2 (42) 78 7 23 46 33 49 14 39 (2) 1843 42 1 13 3 9 4 1 1 1 27 1 (41) 79 6 27 9 18 49 31 54 3 81 (2) 1844 42 1 16 5 3 4 4 1 4 1 1 3 1 2 1 (20) 80 5 16 25 19 17 59 16 4341 20 53 23 61 (4) 1845 42 1 20 (2) 81 4 (41) 1846 42 1 27 1 1 1 8 1 7 1 2 3 2 1 1 3 3 (6) 82 3 55 30 53 9 69 10 55 27 23 30 39 43 22 25 (13) 1847 42 1 (41) 83 (2) 1848 42 (1) (84) 1850 43 (86) (1) 182 THE PELL EQUATION 1851 43 (43) (2) 1852 43 28 1 2 9 4 2 2 1 2 1 7 10 1 11 2 3 4 1 3 57 28 9 19 33 27 49 24 57 11 8 51 33 44 29 24 17 56 3 3 2 11 1 6 3 1 (16) 21 23 36 7 69 12 21 63 (4) 1853 43 21 (1) (1) 4 (43) (43) 1854 43 17 4 1 2 1 1 1 3 1 (8) 5 18 53 25 45 34 47 19 62 (9) 1855 43 14 2 1 8 (1) 6 29 54 9 (70) 1856 43 12 3 2 1 3 (21) 7 25 28 49 23 (4) 1857 43 10 1 3 5 7 1 1 1 4 2 2 1 1 11 1 2 1 2 8 61 21 16 11 51 32 49 17 33 29 37 48 7 59 24 47 31 (28) (3) 1858 43 9 1.1 3 4 1 1 (42) 9 46 39 23 18 39 47 (2) 1859 43 8 1 1 1 1 2 1 5 2 (3) 10 49 35 38 43 25 55 14 35 (22) 1860 43 7 1 (4) 11 64 (15) 1861 43 7 5 1 1 1 1 3 1 2 (2) (2) 12 15 47 36 37 45 20 51 27 (31) (31) 1862 43 6 1 1 1 2 7 (2) 13 49 34 43 31 11 (38) 1863 43 6 6 2 9 7 1 (2) 14 13 38 9 11 58 (23) 1864 43 5 1 2 111 1 2 (10) 15 56 23 60 7 49 36 33 (8) 1865 43 5 2 1 1 (2) (2) 16 31 40 41 (29) (29) 1866 43 5 (14) 17 (6) 1867 43 4 1 3 1 2 1 28 14 2 1 2 1 1 9 (43) 18 57 19 54 23 62 3 6 31 46 27 38 47 9 (2) 1868 43 4 1 1 6 10 1 1 1 7 4 1 (20) 19.41 44 13 8 53 31 52 11 17 67 (4) 1869 43 4 3 4 1 3 1 1 21 17 (4) 20 25 17 57 20 39 47 4 5 (21) 1870 43 4 9 2 1 3 (2) 21 9 30 49 21 (34) THE PELL EQUATION 183 1871 43 3 1 1. 1 1 1 4 2 16 1 5 1 2 2 8 4 2 22 61 7 50 35 37 46 17 38 5 70 13 55 26 35 10 19 34 3 3 5 1 7 (43) 23 25 14 65 11 (2) 1872 43 3 (1) 23 (52) 1873 43 3 1 1 2 7 2 11 1 8 1 2 3 3 1 4 1 1 1 3 24 43 39 32 11 39 7 72 9 56 27 24 21 57 16 49 33 48 19 1 10 28 1 3 6 (2) (2) 63 8 3 64 21 13 (33) (33) 1874 43 3 2 4 1 1 1 (42) 25 34 17 50 31 55 (2) 1875 43 3 3 7 1 1 2 1 (13) 26 25 11 46 39 25 59 (6) 1876 43 3 5 (12) 27 16 (7) 1877 43 3 12 21 1 1 2 (1) (1) 28 7 4 49 37 28 (41) (41) 1878 43 2 1 (42) 29 57 (2) 1879 43 2 1 7 4 1 2 5 2 2 1 3 17 14 2 1 1 3 1 30 53 11 18 51 29 15 34 27 50 23 5 6 33 38 45 19 66 28 8 1 1 1 2 1 4 2 1 2 9 3 1 5 (43) 3 10 51 33 46 25 54 17 30 45 31 9 22 57 15 (2) 1880 43 2 1 (3) 31 49 (20) 1881 43 2 1 2 3 10 1 1 4 1 (8) 32 45 29 25 8 45 41 16 65 (9) 1882 43 2 1 1 (1) (1) 33 41 38 (39) (39) 1883 43 2 1 1 (5) 34 37 47 (14) 1884 43 2 2 7 2 (28) 35 33 11 40 (3) 1885 43 2 (2) 36 (29) 1886 43 2 2 1 (42) 37 25 61 (2) 1887 43 2 3 1 (1) 38 21 47 (34) 1888 43 2 4 1 1 1 1 2 9 3 1 2 (21) 39 17 47 36 39 41 32 9 23 48 31 (4) 1889 43 2 6 5 3 1 1 2 2 3 17 10 1 (4) (4) 40 13 16 23 43 40 29 32 25 5 8 65 (17) (17) 184 THE PELL EQUATION 1890 43 2 9 (6) 41 9 (14) 1891 43 2 16 1 8 1 2 1 1 2 1 1 1 5 6 28 1 4 (1) 42 5 74 9 58 25 42 41 27 45 34 49 15 14 3 70 15 (62) 1892 43 (2) (43) 1893 43 1 1 (28) 44 43 (3) 1894 43 1 1 11 1 13 1 1 2 2 1 1 1 1 9 17 3 3 2 5 45 42 7 75 6 49 37 30 29 42 39 35 50 9 5 26 23 35 15 2 1 2 11 6 8 1 1 4 3 3 1 528 1 5 30 47 27 39 46 13 10 45 41 18 25 21 58 15 3 71 14 3 1 (42) 21 65 (2) 1895 43 1 1 7 2 2 2 1 (16) 46 41 11 34 31 26 59 (5) 1896 43 1 1 5 3 (3) 47 40 15 25 (24) 1897 43 1 1 4 (12) 48 39 19 (7) 1898 43 1 1 (3) (3) 49 38 (23) (23) 1899 43 1 1 2 1 2 1 1 1 3 6 2 3 (43) 50 37 27 49 26 45 35 46 23 13 35 25 (2) 1900 43 1 1 2 3 (4) 51 36 31 24 (19) 1901 43 1 (1) (1) 52 (35) (35) 1902 43 1 1 1 1 2 1 3 (14) 53 34 39 43 26 51 23 (6) 1903 43 1 1 1 1 1 9 14 2 (3) 54 33 43 34 51 9 6 37 (22) 1904 43 1 1 1 2 1 (4) 55 32 47 25 55 (16) 1905 43 1 1 1 4 1 3 1 3 2 1 2 1 (16) 56 31 51 16 59 19 56 21 29 49 24 61 (5) 1906 43 1 1 1 11 1 4 4 1 1 1 (5) (5) 57 30 55 7 66 17 18 49 33 50 (15) (15) 1907 43 1 2 (43) 58 29 (2) 1908 43 1 2 7 1 1 1 1 4 1 5 1 (8) 59 28 11 49 36 37 47 16 63 13 68 (9) 1909 43 1 2 4 28 1 8 1 (2) 60 27 20 3 76 9 60 (23) THE PELL EQUATION 1910 43 1 2 2 1 2 (8) 61 26 29 46 31 (10) 1911 43 1 2 1 1 (28) 62 25 38 49 (3) 1912 43 1 2 1 1 1 9 12 2 1 1 3 4 1 6 2 (10) 63 24 47 33 52 9 7 33 39 44 23 17 63 12 39 (8) 1913 43 1 2 1 4 2 1 1 10 2 1 11 1 4 (1) (1) 64 23 56 17 32 37 49 8 29 56 7 67 16 (43) (43) 1914 43 1 (2) 65 (22) 1915 43 1 3 5 1 1 2 29 3 3 1 5 2 14 7 1 (7) 66 21 15 46 39 29 35 9 26 21 59 14 39 6 11 69 (10) 1916 43 1 3 2 1 1 2 1 10 4 1 1 16 1 (20) 67 20 31 40 43 25 59 8 19 40 47 5 79 (4) 1917 43 1 3 1 1 1 1 1 (2) 68 19 47 36 41 37 44 (27) 1918 43 1 3 1 7 (6) 69 18 63 11 (14) 1919 43 1 4 6 17 2 1 3 3 (4) 70 17 14 5 31 49 22 25 (19) 1920 43 1 4 2 (21) 71 16 39 (4) 1921 43 1 4 1 5 1 10 9 1 1 1 5 5 (3) (3) 72 15 64 13 69 89 53 32 51 15 16 (25) (25) 1922 43 1 5 3 1 1 1 4 1 1 11 1 (42) 73 14 23 47 34 49 17 41 46 7 79 (2) 1923 43 1 5 1 3 7 1 2 2 (43) 74 13 59 22 11 57 26 37 (2) 1924 43 1 6 39 2 3 28 1 (20) 75 12 27 9 36 25 -3 81 (4) 1925 43 1 (6) 76 (11) 1926 43 1 7 1 3 1 2 1 2 1 1 16 1 (42) 77 10 63 19 54 25 50 27 38 49 5 81 (2) 1927 43 1 8 1 3 3 1 1 3 1 1 1 1 2 4 2 (43) 78 9 62 21 23 42 43 21 46 37 39 42 31 18 39 (2) 1928 43 1 (9) 79 (8) 1929 43 1 11 I 1 3 1 1 1 (28) 80 7 47 40 21 49 32 55 (3) 1930 43 1 13 1 1 1 (9) (9) 81 6 55 31 54 (9) (9) 1931 43 1 16 1 1 2 3 8 2 (43) 82 5 50 37 31 25 10 41 (2) 185 186 THE PELL EQUATION 1932 43 1 (20) 83 (4) 1933 43 1 28 3 9 2 3 1 2 2 12 7 4 21 1 2 1 6 1 1 84 3 28 9 37 21 52 27 36 7 12 21 4 63 23 59 12 47 39 2 1 1 1 1 (6) (6) 28 43 39 36 49 (13) (13) 1934 43 1 (42) 85 (2) 1935 43 (1) (86) 1937 44 (88) (1) 1938 44 (44) (2) 1939 44 29 2 1 9 8 1 2 2 1 2 4 3 1 3 2 3 (12) 3 30 55 9 10 57 27 29 47 30 19 22 55 21 34 25 (7) 1940 44 (22) (4) 1941 44 17 1 1 1 1 2 1 11 1 6 2 2 1 2 7 1 1 1 3 5 52 35 39 44 25 60 7 71 12 35 28 47 31 11 52 33 49 20 1 3 21 1 3 4 6 1 1 5 2 1 (28) 55 23 4 65 21 20 13 44 43 15 28 59 (3) 1942 44 14 1 2 9 2 4 1 2 2 4 4 1 2 29 (44) 6 57 29 9 38 17 54 27 34 19 18 51 31 3 (2) 1943 44 12 1 1 (2) 7 49 38 (29) 1944 44 (11) (8) 1945 44 9 1 3 1 2 1 7 3 1 1 4 1 (16) 9 64 19 55 24 59 11 24 41 45 16 69 (5) 1946 44 8 1 4 3 3 (12) 10 65 17 25 26 (7) 1947 44 (8) (11) 1948 44 7 2 1 9 7 1 11 1 2 1 3 10 1 3 3 2 2 1 2 12 29 56 9 11 72 7 61 24 53 23 8 63'21i24 33 28 49 27 1 1 3 1 1 1 2 29 (22) 41 44 21 47 36 43 33 3 (4) 1949 44 6 1 3 (1) (1) 13 61 20 (43) (43) 1950 44 6 3 2 1 (2) 14 25 29 49 (26) 1951 44 5 1 7 5 14 1 1 8 3 6 29 2 1 5 4 1 2 1 15 66 11 17 6 45 43 10 27 13 39 9 30 53 15 18 55 25 51 THE PELL EQUATION 187 21 1 1 17 29 2 1 1 3 2 2 1 1 (43) 26 47 33 54 5 3 34 39 45 22 33 30 37 51 (2) 1952 44 5 1 1 (21) 16 41 47 (4) 1953 44 5 5 3 (12) 17 16 27 (7) 1954 44 4 1 9 (44) 18 65 9 (2) 1955 44 4 1 (1) 19 49 (34) 1956 44 4 2 2 3 7 1 2 1 (28) 20 33 32 25 11 60 23 64 (3) 1957 44 4 4 1 21 3 (4) 21 17 69 4 27 (19) 1958 44 (4) (22) 1959 44 3 1 5 6 1 1 1 2 1 8 7 1 (13) 23 58 15 13 51 34 47 25 59 10 11 73 (6) 1960 44 3 1 2 9 (2) 24 49 31 9 (40) 1961 44 3 1 1 7 2 12 5 (2) 25 40 47 11 40 7 16 (37) 1962 44 3 2 1 1 9 3 1 11 1 (8) 26 31 38 49 9 22 63 7 74 (9) 1963 44 3 3 1 2 3 2 14 1 1 4 2 2 (6) 27 22 51 29 23 38 6 47 41 18 33 34 (13) 1964 44 3 6 2 17 3 1 3 1 10 3 2 4 1 3 1 1 1 1 1 28 13 40 5 23 56 19 65 8 25'35 17 59 20 47 37 40 41 35 1 12 (22) 52 7 (4) 1965 44 3 21 1 (4) 29 4 71 (15) 1966 44 2 1 17 14 1 2 1 1 1 1 2 2 1 9 6 1 2 1 30 57 5 6 61 25 45 38 39 43 30 27 58 9 13 57 26 41 1 4 2 1 5 4 2 29 8 1 5 (44) 45 18 29 54 15 19 39 3 10 67 15 (2) 1967 44 2 1 (5) 31 53 (14) 1968 44 2 1 3 (5) 32 49 23 (16) 1969 44 2 1 2 9 2 17 3 1 1 1 3 1 1 2 3 3 3 1 11 33 45 32 9 41 5 24 47 35 48 21 45 40.31 24 25 21 64 7 1 10 5 1 4 1 2 2 5 2 29 (8) 75 8 15 63 16 55 27 35 15 40 3 (11) 188 THE PELL EQUATION 1970 44 2 (1) (1) 34 (41) (41) 1971 44 2 1 1 9 3 (1) 35 37 50 9 23 (54) 1972 44 2 2 5 6 1 1 1 4 1 9 (22) 36 33 16 13 52 33 51 16 67 9 (4) 1973 44 2 2 1 1 (3) (3) 37 29 41 44 (23) (23) 1974 44 2 3 17 (2) 38 25 5 (42) 1975 44 2 3 1 (2) 39 21 54 (25) 1976 44 2 4 1 2 (1) 40 17 56 25 (52) 1977 44 2 6 2 1 10 2 3 4 2 1 1 4 1 (28) 41 13 29 57 8 37 24 19 32 39 47 16 71 (3) 1978 44 2 9 2 1 1 2 3 1 1 1 12 14 1 (2) 42 9 33 41 42 31 22 49 33 54 7 6 63 (23) 1979 44 2 17 3 2 1 2 1 6 8 1 2 1 (43) 43 5 26 29 50 25 58 13 10 61 23 65 (2) 1980 44 (2) (44) 1981 44 11 29 5 19 17 1 2 2 1 5 4 3 1 1 1 2 2 45 44 3 15 68 9 5 60 27 28 55 15 20 23 47 36 45 29 33 3 7 7 1 21 2 1 1 1 6 1 3 1 4 2 31 25 12 11 75 4 33 44 35 51 12 63 19 60 17 36 21 57 3 1 2 (12) 20 51 31 (7) 1982 44 1 1 12 4 1 1 1 1 5 1 3 (44) 46 43 719 47 38 37 49 14 59 23 (2) 1983 44 1 1 7 1 1 2 5 1 (28) 47 42 11 49 38 33 14 73 (3) 1984 44 1 1 5 2 3 2 (2) 48 41 15 36 23 33 (31) 1985 44 1 1 4 5 2 1 7 2 (2) (2) 49 40 19 16 29 56 11 35 (31) (31) 1986 44 1 1 3 2 1 2 5 1 1 3 (44) 50 39 23 30 47 31 15 46 41 25 (2) 1987 44 1 1 2 1 3 1 (43) 51 38 27 54 19 69 (2) 1988 44 1 1 2 2 1 2 (12) 52 37 31 29 47 32 (7) 1989 44 1 1 221 1 (8) 53 36 35 4 77 (9) THE PELL EQUATION 189 1990 44 1 1 1 1 3 1 1 1 5 3 3 1 14 9 1 5 2 (8) 54 35 39 46 21 49 34 51 15 26 21 65 6 9 69 14 39 (10) 1991 44 1 1 1 1 1 2 1 17 (8) 55 34 43 37 46 25 62 5 (11) 1992 44 1 1 1 2 1 1 (10) 56 33 47 28 39 49 (8) 1993 44 11 1 429 1 1 4 1 2 1 9 5 2 10 1 2 2 1 57 32 51 19 3 48 41 17 57 24 61 9 16 39 8 59 27 31 39 1 6 3 (1) (1) 48 13 24 (43) (43) 1994 44 1 1 1 8 3 1 3 (3) (3) 58 31 55 10 23 55 22 (25) (25) 1995 44 1 (1) 59 (30) 1996 44 1 2 10 1 5 (22) 60 29 8 69 15 (4) 1997 44 1 2 4 1 11 1 21 2 2 1 (2) (2) 61 28 17 68 7 79 4 37 28 49 (29) (29) 1998 44 1 2 3 9 1 1 1 (2) 62 27 26 9 53 34 47 (27) 1999 44 1 2 2 4 1 1 5 1 5 8 1 3 2 1 2 1 1 1 63 26 35 18 43 45 14 65 15 10 63 21 30 49 27 45 38 41 1 1 1 1 14 3 1 1 29 4 4 2 5 1 1 17 2 39 42 35 53 6 25 39 50 3 21 19 37 15 42 47 5 30 1 12 9 1 5 1 (43) 57 79 70 13 75 (2) 2000 44 1 2 1 1 2 3 (5) 64 25 44 41 31 25 (16) 2001 44 1 2 1 2 1 4 1 6 17 1 (2) 65 24 53 25 57 16 65 13 5 64 (23) 2002 44 1 2 1 9 5 (6) 66 23 62 9 17 (14) 2003 44 1 3 12 1 1 6 2 1 2 1 3 6 7 1 (43) 67 22 746 43 13 31 49 26 53 23 14 11 73 (2) 2004 44 1 3 3 1 1 1 3 1 5 5 2 2 1 2 1 (6) 68 21 23 48 35 49 20 61 15 16 35 28 51 25 59 (12) 2005 44 1 3 221 1 (16) 69 20 39 481 (5) 2006 44 1 3 1 2 1 1 1 (4) 70 19 55 26 47 35 50 (17) 2007 44 1 (3) 71 (18) 2008 44 1 4 3 1 1 6 1 9 (11) 72 17 24 41 47 12 71 9 (8) 190 THE PELL EQUATION 2009 44 1 4 1 I 1 1 2 3 1 1 17 2 1 2 1 10 (2) 73 16 49 37 40 43 31 23 40 49 5 32 49 25 61 8 (41) 2010 44 1 (4) 74 (15) 2011 44 1 5 2 2 1 1 8 2 1 1 1 1 29 3 1 1 4 6 75 1353039493342414 35 30 39 49 10 33 42 41 35 54 3 25 42 45 19 13 1271491 3 5 1 21111 31 55 30 11 65 18 9 63 22 15 57 26 45 39 38 47 21 50 1 1 14 3 3 1 17 5 1 11 1 (43) 33 55 627 21 66 5 15 70 7 81 (2) 2012 44 1 5 1 10 2 1 4 (22) 76 13 71 831 52 19 (4) INDEX OF NAMES. Numbers refer to pages. Abul Wefa, 39. Alkacadi, 41. Alkarkhi, 30, 39, 40, 45. Amthor, M., 21. Apastamba, 6. Archimedes, 15, 16, 19, 41, 45, 63, 128, 130. Arndt, F., 115, 116. Aryabhatta, 26. Aubel, H. van, 131. Aubry, A., 60, 86, 150, 160. Auric, 158. Bachet, 31, 58, 71, 72. Bachman, P., 85, 136, 141. Barlow, P., 62, 97, 114, 123, 133. Baudhayana, 6, 8, 127. Beligne, A., 138. Bell, A. H., 21, 147, 149, 153, 155, 158. Berkhan, 143. Besouclein, 146. Bhaskara, 9, 27, 29, 31, 33, 36, 37, 38, 49, 152. Bickmore, C. E., 97, 135. Bills, S., 123. Blaikie, J., 153. Booth, 123. Bortolotti, E., 136. Boutin, A., 139, 140, 144, 148, 159. Brahmagupta, 9, 26, 27, 29, 30, 31, 147, 152. Brancker, T., 2. Bricard, R., 161. Brouncker, W., 1, 2, 32, 47, 49, 50, 51, 52, 56, 57, 58, 59, 64, 70, 71, 76, 152. Brocard, H., 125, 130, 138, 140, 144, 147, 154. Birk, A., 6. Buteo, J., 45, 46. Cahen, E., 142, 159. Calzolari, L., 120. Candido, G., 152. Cantor, M., 4, 8, 9, 10, 13, 69. Cardan, 45. Catalan, E., 97, 118, 129, 140. Cataldi, 45. Cayley, A., 97, 117, 135. Chatelet, A., 160. Christie, R. W. D., 142, 147, 148, 149, 152, 153, 154, 155, 156, 158, 160. Chrystal, 73. Chuquet, N., 42, 44, 46, 132. Colebrooke, H. T., 27, 30, 31, 32, 34, 37, 38. Collins, M., 122. Cornacchia, G., 158. Cosali, P., 113. Crishna-bhatta, 32. Crowell, E. B., 152. Cunningham, A., 1, 95, 96, 97, 142, 147, 149, 152, 154, 155, 156, 157, 158, 159, 160. Curtze, M., 138. d'Andrea, C., 115. Davis, R. F., 147. de Billy, J., 46. Dedekind, 81, 86. Degen, C. F., 95, 96. De Lagny, 16. de la Roche, 43, 44. de Ortega, J., 44. d'Ocagne, M., 130. Didon, F., 121. Digby, 49, 50, 56, 57. Diophantus, 22, 23, 24, 25, 26, 30, 31, 39, 46, 47, 48, 49, 50, 128, 152. Dirichlet, G. L., 79, 80, 81, 82, 85, 86, 88, 115, 132, 133, 141. Dubouis, 159. Dupuis, J., 13. Durfee, W., 129. Egen, P. N. C., 114. El-Hassar, 40. Elphinstone, M., 152. Emmerich, 140. Enestr6m, G., 2. Epstein, 151. Escott, E. B., 140, 144, 146, 148, 151, 159, 161. Euclid, 48. 191 192 INDEX OF NAMES Euler, L., 1, 2, 31, 52, 59, 60, 61, 62, 63, 64, 65, 66, 67, 70, 71, 72, 73, 76, 85, 95, 113, 147, 149, 154. Evans, A. B., 120, 123, 126. Even, 130. Fermat, P., 1, 19, 46j 47, 48, 49, 50, 51, 52, 55, 56, 57, 58, 59, 70, 71, 72, 76, 141, 143. Frontene, G., 159, 160. Forsyth, A., 135. Frattini, G., 131, 133, 134, 137, 143, 148, 149, 160. Frenicle, 49, 57, 58. Fricke, R., 138. Friedlein, G., 9. Frischauf, J., 119. Fuss, P. H., 1, 59, 63. Gauss, C. F., 1, 71, 76, 78, 81, 89, 117, 141. Geminos, 14. Gerono, G. C., 117. Gill, C., 121. Goldbach, C., 1, 59, 63. Gopel, A., 116. Goulard, 140. Gow, J., 21. Giunther, S., 4, 6, 8, 14, 16, 20, 22, 42. Hart, 121, 122, 123, 125. Hart, D. S., 124, 143. Hart, G., 123. Hankel, H., 2, 26, 31. Hayashi, T., 161. Heath, T. L., 15, 18, 20, 22, 23, 58, 60. Heilermann, 17. Hewlett, J., 52, 62, 76. Hilprecht, H. V., 4, 5. Hochheim, A., 39. Hoernle, 30, 31. Hoiiel, 91. Hoffman, K. E., 126. Holm, A., 146, 152, 154. Hopkins, G. H., 121. Hultsch, F., 9, 10, 11, 14, 18, 21. Hunrath, 18. Hurwitz, A., 133. Hypsicles, 128. Jacobi, C. G. J., 88, 115. Jamet, 130. Jonquieres, E. de, 124, 129, 140, 141. Junge, G.,. 5 Kaye, G. R., 26, 30. Khanikof, N. de, 120. Klein, F., 138. Kommerell, K., 150. Konen, H., 2, 52, 68, 143. Konig, K. F., 117. Kramp, 113. Krishnamachari, B., 152. Kroll, 10. Kronecker, M., 1, 81. Krumbiegel, B., 21. Kunerth, A., 124, 126, 128. Lagrange, J. L., 31, 55, 70, 71, 72, 73, 74, 75, 76, 85, 95, 113, 117, 129, 131, 132, 143, 152. Laisant, A., 120. Landry, F., 117. Legendre, A. M., 1, 70, 79, 80, 95, 96, 115, 132. Lemoine, E., 135. Lessing, 21. Leudesdorf, C., 126. Levy, A., 157, 160. Lionet, 127. Lodge, A., 135. Longchamps, G. de, 136, 140, 146. Lucas, E., 139. Luce, J. B., 116. MacMahon, P. A., 157. Mahler, E., 130. Maillet, E., 136. Malebranche, 58. Malo, 154, 155. Marcolongo, R., 132. Marre, A., 43. Martin, A., 120, 123, 124, 133, 134, 140, 143. Mascart, J., 10. Mathews, G. B., 93. Matthiessen, L., 121. Mayers, F. N., 153. Meissel, E., 143. Merriman, M., 21. Meyer, A., 132, 143. Minding, E. F. A., 114. Minnigerode, B., 121, 131. Moret-Blanc, 120. Moreau, C., 121. Muller, R., 132. Naber, H. A., 4. Niewenglowski, B., 157. Nesselmann, G. H. F., 13, 20, 23. Netto, E., 73. Newton, I., 127. Nash, A. M., 122, 126. Oppermann, 18. Orchard, H. L., 132. INDEX OF NAMES 193 Ottinger, L., 119. Ozanam, 58, 76. Paciuolo, 45. Pal, J. C., 160. Palletti, G., 114. Palmstr6m, A., 137, 138, 139, 141. Pappus, 24, 128. Pepin, T., 1, 127, 129. Pell, J., 1, 2, 19, 59. Perott, J., 45, 132. Pezzi, F., 113. Picard, E., 130. Pisani, F., 127. Pistor, 143. Plamenewshy, H., 132. Planudes, M., 41. Plato, 9, 128. Polignac, de, 142. Poussin, C. J. de la V., 137. Proclus, 10, 11, 13, 14. Puisse, 141. Pythagoras, 4, 15, 22. Rahn, J. H., 2. Ranucci, D. N., 161. Rau, B. H., 132. Realis, S., 127. Reid, L. W., 161. Rhabdas, N., 45. Ricalde, G., 143, 144. Richaud, C., 118. Richaud, H., 97, 131. Roberts, S., 97, 125, 126, 128, 130, 134. Rocquigny, G. de, 130, 139. Rodet, M. L., 5, 18, 42, 43, 127. Romero, J., 145. Rudis, 146, 151, 154. Sandier, J., 151. Schaewen, P. von, 46. Schlomilch, 0., 121. Schmidt, W., 122, 143. Sch6nborn, 18. Schroder, J., 4, 151. Schwering, K., 135. Seeling, P., 119, 120. Serret, J. A., 141. Sharpe, J. W., 126. Simon, M., 6. Smith, D. E., 5. Smith, H. J. S., 52, 70, 118, 123. Sommer, J., 92, 157. Speckman, G., 93, 94, 136, 137, 143. Stern, M. A., 114, 117, 119, 131. St6rmer, C., 138, 139, 140. Strachey, E., 31, 37, 38, 39. Struve, J., 20, 21. Struve, K. L., 20. Stuart, T., 158. Surin, 34. Suter, H., 39, 40. Sylvester, J. J., 135. Tanner, H. W. L., 137. Tannery, P., 6, 7, 9, 15, 16, 18, 19, 24, 42, 45, 63, 128, 132, 146. Tano, F., 133. Tartaglia, 45. Tchebicheff, P., 116. Tebay, S., 122. Tedenat, 113. Teilhet, P. F., 147, 148. Tenner, G. W., 68, 69, 143. Theon of Smyrna, 13, 14, 15, 17, 129. Thibaut, G., 5, 6, 7. Treutlein, 42. Thielman, von, 155. Thue, A., 139. Todhunter, I., 66. Vahlen, K. H., 159. Vieta, 47. Vogt, H., 5. Volpicelli, P., 116. Wallis, J., 1, 2, 49, 50, 55, 56, 57, 58, 59, 62, 71, 72, 141. Wantzel, L., 116. Waschke, H., 41. Weber, H., 135. Weill, 131. Weissenborn, H., 129, 130. Werebrusow, A. S., 145, 146, 149. Wertheim, G., 2, 49, 141. Woepcke, F., 39, 41. Zeuthen, H. G., 7, 10, 15, 16, 30, 63.