t 



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



where C-=f 2 A 2 +2fgAB-g 2 B 2 = g 2 A 2 +2f 2 B 2 . Now / divides r, g divides s, 

 while r and s are < Va;. Hence each set of integral solutions of (2) with 

 x 2 =|=1 leads to a set of smaller solutions. For/=13, g = l, ft =239, we 

 get A = 3-113, B= -2-113 or 3-84; for the first, cr = 113, r = 39, s=-2, 

 a; = 1525, ?/=-1343; for the second, o- = 3, r = 13-113, s = 84, z = 2165017, 

 y=- 2372159. 



Lebesgue noted that x*2 m y*=z 2 has integral solutions only when 

 w = 4n3 and then may be made to depend upon (2) ; likewise, 2 m x* y*=z 2 

 only when w = 4n+l. But 4 d=2/ 4 = 2 m 2 2 is impossible in integers. All of 

 these cases except x*Sy* = z 2 and 8z 4 y* = z 2 had been treated by Euler, 

 Algebra 2, Ch. 13, whose 49 solution of z 4 2y* = z 2 is incomplete (Art. 211). 



E. Lucas 57 gave a complete solution of z 4 2y* = z 2 and x*+Sy* = z 2 , 

 based on the complete solution of u 2 +v 2 = y*. He 46 obtained the usual 

 results concerning Fermat's 37 problem. 



T. Pepin 58 treated 2z 4 -l = D by his 157 final method. He 59 remarked 

 that Lebesgue 56 merely stated, but did not prove, that his formulas lead to all 

 solutions of (2) under a given limit. Pepin obtained the same solutions 

 by a simpler method proved complete. If x, y, z are relatively prime by 

 pairs, 



x = p 2 +q 2 , zyH=(l+i)(p+qi)\ 



where p, q are relatively prime and q may be taken even. Then 



the lower sign being excluded by use of modulus 8. Thus 



* 



(p 2 -q 2 +2pq)y = 2r 2 , (p 2 -q 2 +2pq)^Fy = 4s 2 , rs = pq, 



r, s being relatively prime. By the last, p = Xju, q = hk, r = \h, s = /jLk, where 

 X, /z, h, k are integers relatively prime by pairs, k alone being even. From 

 p 2 q 2 -\-2pq = r 2 + 2s 2 (the lower sign having been excluded by modulus 4), 

 -2\iJihk+\ 2 (h 2 -fji 2 ) =0, whence 



\ 2 



Thus M, h form a solution of (2), while X 4 2k 2 = D. The above is valid if 

 x>\, whence <?=t=0. Thus any solution except x = y z = l leads to a 

 solution X' = IJL, y' = h, z' = ^2^ h 4 , in smaller numbers, and given by 



x = \ 2 2 +h 2 k 2 , ?/ = \ 2 h 2 - 2fj. 2 k 2 , z = y 2 -S\hk(\ 2 2 - h 2 k 2 ), 

 where 2n 4 h 4 = t-, A;/X = (ju/idbO/(2M 2 +ft 2 ), from whose numerator and de- 



nominator common factors are to be suppressed. We can therefore com- 

 pute the successive sets of solutions of (2) starting with x = y = z = 1 . 



67 Recherches sur 1'analyse indtermin6e, Moulins, 1873, 25-32. Extract from Bull. Soci6t6 



d'Emulation Dept. de 1'Allier, 12, 1873, 467-72. Same in Bull. Bibl. Storia Sc. Mat. Fis., 

 10, 1877, 239-45. 



68 Atti Accad. Pont. Nuovi Lincei, 30, 1876-7, 220-2. 



69 Ibid., 36, 1882-3, 37^0. 



