CHAP. XII] PELL EQUATION, ax z +bx-}-c= D. 345 



view of J. L. Heiberg, 18 P. Tannery, 19 F. Hultsch, 20 T. L. Heath, 21 and 

 S. Giinther. 22 



A. H. Bell 23 found a " complete solution," based on the an 2 + 1 = D of 

 Leiste, involving numbers of 206545 digits, as by Amthor. 16 



G. Loria, 23a M. Merriman, 236 and R. C. Archibald 230 gave accounts of the 

 cattle problem. 



Diophantus (about 250 A.D.) was frequently led to special Pell equa- 

 tions in solving problems in his Arithmetica. In II, 12, 13, 14, 29, he 

 made y z + 1, y z + 12, y* - - 1, y* + 1, 9?/ + 9 equal to a square z 2 , by 

 taking z = y 4, y 4, y 2, y 2, Zy - - 4, respectively, and similarly 

 in II, 30. In III, 12, 13, he avoided the initial equations 52z 2 + 12 = D, 

 266z 2 10 = D, since 52 and 266 are not squares [though x = 1 is 

 a solution of each], and, beginning anew, was led to y 2 + 12 = D, 

 77 2 z 2 160 = D, which he solved by equating them to (y + 3) 2 and 

 (77s - 2) 2 , respectively. In IV, 8, 33, he treated 2x 2 + 4 = D = (2x - 2) 2 

 and 7m 2 + 81 = D = (8ra + 9) 2 . In V, 12, 14, he discussed 



26z 2 + 1 = D = (5z + I) 2 



and 30 2 + 1 = D = (5x + I) 2 . So far, the problems solved are all of 

 the form ax 2 + b = D with either a or b a square. In VI, 12, he stated the 

 lemma: Given two numbers whose sum is a square, we can find an infinitude 

 of squares s such that, when the square s is multiplied by one of the given 

 numbers and the product is added to the other, the result is a square. 

 Thus, given the numbers 3 and 6, let s = (x + I) 2 ; then shall 



3(z + I) 2 + 6 = 3x 2 + 6x + 9 = D, 



say (3 3x) 2 , whence x = 4; and an infinitude of other solutions can be 

 found. This lemma is applied in VI, 13, 14 to 12z 2 + 24 = D to obtain 

 the solutions x = 1, 5. In VI, 15, 15x 2 - - 36 = D is said to be impossible 

 since 15 is not a sum of two squares. In VI, 16, he made the important 

 statement that, given one solution of Ax 2 B = y 2 , we can find a second 

 solution; thus, given 3-5 2 - 11 = 8 2 , set x = 5 + z, whence 



3(5 + z) 2 - 11 = 3z 2 + 302 + 8 2 

 will be the square of 8 2z for z = 62. In VI, 12, he had made the more 



18 Questiones Archimedeae, Diss. Hauniae, 1879, 25-27; Philologus, 43, 1884, 486. 



19 Mem. soc. sc. phys. nat. Bordeaux, (2), 3, 1880, 370; Bull, des Sc. Math, et Astr., (2), 5, I, 



1881, 25-30; Bibl. Math., 3, 1902, 174. Reprinted in Tannery's Memoires scientifiques, 

 1, 1912, 103-5, 118-23. 



20 Archimedes, in Pauly-Wissowa's Real-Encyclopadie, Hi, 1896, 534, 1110. 



21 Diophantus, ed. 2, 1900, 11-12, 122, 279; Archimedes, 1897,319; Archimedes' Werke, 1914, 



471-7. 



22 Die quadr. Irrationalitaten, etc., Zeitschrift Math. Phys., Abh. Gesch. Math., 27, 1882, 92. 



This and K. Hunrath's Ueber das Ausziehen der Quadratwurzel bei Griechen und Indern, 

 1883, were reviewed in La Revue Scientifique, 1884, I, 81-3, 499-502. 



23 Math. Magazine, Washington, 2, 1895, 163^; Amer. Math. Monthly, 2, 1895, 140-1 



(1, 1894, 240). 



230 Le scienze esatte nell'antica Grecia, ed. 2, 1914, 932-9. 

 ^ The Popular Science Monthly, 67, 1905, 660-5. 

 23c Amer. Math. Monthly, 25, 1918, 411-4. 



