624 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxn 



Let s, t be relatively prime solutions of (7) . Then s is odd and 

 w+s 2 = 2p.w, u s- = 2fj,p, 8 4 = 4ju 2 wp, 



where co and p are relatively prime. Thus /* divides i 2 . Also S 2 = ju(w p). 

 Hence ju = 1 , co = 2q 4 , p = r 4 , or co = q 4 , p = 2r 4 , whence 



w = 2g 4 +r 4 , s 2 = 2? 4 -r 4 ; or u = q*+2r\ s 2 = g 4 -2r 4 . 



Conversely, if 2^ 4 r 4 = s 2 or <? 4 2r 4 = s 2 and we set t = qr, we have solutions 

 s, t of (7). If s and t are relatively prime and numerically distinct from 

 unity, the same is true of q and r, while the greater of s, t exceeds the greater 

 of q, r. The first of these two equations is of type (2) . 



Applying to the second, g 4 2r 4 = s 2 , a discussion entirely similar to that 

 just used, Lagrange obtained 



s = 8n 4 -p 4 , g 2 = 8n 4 -hp 4 ; or s = n 4 -8p 4 , g 2 = n 4 +8p 4 . 



The former becomes the latter if we interchange n with p and change the 

 sign of s. The solution of q 4 2r 4 = s 2 is therefore reduced to that of 

 2 2 = n 4 +8p 4 , of type (7), by setting r = 2pn, s = n 4 8p 4 . Further, q and r 

 exceed n and p. 



The method leads to all solutions not only of (2) but also of (7) and of 

 ^4 _ 2r 4 =D. Starting with the evident solutions s = t = l, ii = 3 of (7), 

 we deduce the solutions r = 2st = 2, q = u=3, k = 7, of g 4 -2r 4 = & 2 ; and, 

 by (6), (5), solutions of (2): m = 0, n= 1, 1 = 7, x = y = z = l, or ra= 6, 

 w=-7, Z = l, 3 = 13, ?/ = l, 2 = 239. For r = 2, q = 3, s = 7, we deduce the 

 solutions s = 7, = #r = 6, w = 113 of (7); from 13, 1, 239, we get the solutions 

 s = 239, t= 13, u = 57123 of (7). Starting again with one of the latter sets, 

 we obtain new sets of solutions of (2) and g 4 2r 4 = D. In this manner, 

 the sets of solutions of (2) in order of magnitude are (x, y, ) = (!, 1, 1), 

 (13, 1, 239), (1525, 1343, 2750257), (2165017, 2372159, 1560590745759), 

 .... The corresponding sets (3) are p, q = l, 0; 120, -119; 2276953, 

 -473304; and the last two numbers (1). Lagrange therefore proved 

 Fermat's assertion that (1) gives the sides of the least right triangle whose 

 hypotenuse and sum of legs are squares. But Lagrange evidently merely 

 transcribed the statement by Fermat, without making a numerical veri- 

 fication, as the value 15---9of z given by Lagrange (pp. 142, 150, 151; 

 Oeuvres, 380, 393-4) is erroneous [Genocchi 44 ], the correct value being the 

 difference 350- -1 of the last two numbers (1). 



Three of Euler's 55 posthumous papers of 1780 relate to Fermat's 37 

 problem. In the first paper we find a slight modification of his 48 discussion. 

 Taking s = 2, r = 3+v, we get 



if v = 1343/42. Thus p = 1385 -1553, q = 168 -1469, yielding Fermat's solu- 

 tion (1). 



Euler, in the second paper, employed his 48 notations, and obtained 

 = A 2 -2 2 ,whereA=r 2 +2rs-s 2 , = 2rs. Taking A = 2 +2u 2 , B = 2tu, 



66 M6m. Acad. So. St. P6tersbourg, 9, 1819-20, 3; 10, 1821-22, 3; 11, 1830, 1 ; Comm. Arith., 

 II, 397, 403, 421. 



