268 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VII 



S. R6alis 57 gave expressions involving five parameters satisfying 



X 2 + Y 2 + Z 2 = k(x 2 + ?/ 2 + z 2 ) 



for k = 7, 19, 67, and formulas to deduce solutions from a given one. 



L. Kronecker 58 employed the number of classes of bilinear forms in two 

 pairs of cogredient variables to find the number of ways any integer is a SI, 

 in accord with Gauss. 20 



E. Catalan 59 stated that all solutions of x 2 + y 2 = u 2 + v 2 + w 2 are given 

 without repetition by u = x + a, v = y /3, x = sp + (3d, y = sq + ad, 

 where 2s = a 2 + /3 2 + w 2 and a, /3 are relatively prime integers, while 

 Pq - ap = 1. If r, s = a + V a 2 + ft 2 , and n > 1, then 60 



is a G3 and SI. Hence the same is true of x 4n x 4n ~ 2 y 2 + - + y* n for x, y 

 relatively prime integers > 1. 



G. C. Gerono 61 noted that if N 2 is a sum of squares of two consecutive 

 integers, N is a sum of squares of three integers of which two are consecu- 

 tive, as 29 2 = 20 2 + 21 2 , 29 = 2 2 + 3 2 + 4 2 . 



Catalan 62 noted that every power of a SO is a 02 since 



(x 2 + y 2 + z 2 ) 3 = 2/ 2 (3z 2 - x 2 - y 2 ) 2 + z 2 (3z 2 - x 2 - y 2 ) 2 + z 2 (z 2 - 3z 2 - S?/ 2 ) 2 . 



To solve (p. 103) x* + y z = u 2 + w 2 + w 2 , set w = x + a, v = y ft. 

 Then (3y ax = s, where s = |(a 2 + /3 2 + w> 2 ). Take a, /5 relatively prime 

 and w such that s is an integer. For (3q ap = 1, all solutions are given 

 without repetition by x = sp + (38, y = sq 4- # [Catalan 59 ]. 



Catalan stated and E. Fauquembergue 63 proved that, unless x = 1 or 

 4a 2 + 1, (a 2 + l)z 2 = y 2 + 1 implies that a; is a GS, since all solutions (if 

 any) of y 2 Ax 2 = 1 are given by the terms of convergents of even 

 rank in the continued fraction for VA. The latter proved (p. 346) that 

 x 2 + y 2 = u 2 + v z + 1 is satisfied by 2a + 1, a 1, a + 1, 2a and by 

 2a 2 + 1, /3 2 - 1, 2a 2 - 2 + 1, 2a0. 



J. W. L. Glaisher 64 proved that, if the number of representations of 

 8n + 1 by 



(2p + I) 2 + (4r) 2 + (4s) 2 , (2p + I) 2 + (4r + 2) 2 + (4s + 2) 2 

 is Ri, R 2 , respectively, then RI = R 2 unless Sn + 1 is a square, while if 



67 Mathesis, 2, 1882, 64-7. 



58 Abh. Akad. Berlin (Math.), 2, 1883, 52; Werke, 2, 1897, 483. 



69 Assoc. fran. av. sc., 12, 1883, 98-101. 



60 Also stated Nouv. Ann. Math., (3), 3, 1884, 342; Mathesis, 6, 1886, 65, 113. 



61 Nouv. Ann. Math., (3), 2, 1883, 329. 



62 Atti Accad. Pont. Nuovi Lincei, 37, 1883-4, 54-6. 



83 Nouv. Ann. Math., (3), 3, 1884, 538. Cf. Catalan" 1 of Ch. XII. 



84 Quar. Jour. Math., 20, 1885, 94. 



