294 HISTORY OF THE THEORY OF NUMBERS. [CHAP, vni 



Hence, by a change of notation, 



(/ 2 + 20 2 + h z Y = (/ 2 - <7 2 ) 2 + (/ + 0) 2 (<7 + /O 2 



+ (/ + <7) 2 (<7 - hy + (^ - 2/0 + 2 ) 2 . 



Since every odd number is of the form / 2 + 2g z + h?, every odd square is a 

 sum of four squares. 



S. Re"alis 61 used (1) to show that, for any integer p, 



p* = P + Q* + fi + 5 s , 2p + P + Q + R + S = D, 



and that we can find four integers whose algebraic sum is p and the sum of 

 whose squares is p 2 . 



Catalan 62 gave the identity 



Za 2 S(6 7 - c/3) 2 2/ 2 



= (Za/Zaa - 2a/2a 2 ) 2 + {aZ/(6 7 - c/3) + (67 - c|S)Za/ } 2 



+ {62/(&7 - eft) + (ca - a 7 )2a/ } 2 + {c2/(6 7 - C|8) + (aft - ba)2af } 2 , 



expressing a product of three G2 as a ffl . 



Re"alis 63 noted that, for every odd integer p, 



p = P + Q + R + S, p 2 = P 2 + Q 2 + # 2 + S 2 , 

 the algebraic sum of three of P, , S being a square. For, 



p = z 2 + ?/ 2 + 2z 2 

 = (x + s)(x - 2) + (x + z)(z + y) + ( x + z}(z - y) + (2/ 2 + z 2 - 2a). 



Also, if p = 4w + 1, 4n + 2 or 8w + 3, we can make P + Q + R + 3S= D. 

 For, 



p = CS = (a; 2 - 2/2) + (?/ 2 - xz) + (z z - xy) + (xy + xz + yz). 



G. Torelli 64 proved by means of Jacobi's 25 theorem the result (I) that 

 if 2n 1 is not divisible by 3 and if p, q are respectively the numbers of 

 sets of distinct odd integral solutions, not all divisible by 3, of 



2z 2 + ?/ 2 + z 2 = 36(27i - 1), x 2 + y 2 + 2 2 + t 2 = 36(2n - 1), 



then p + 2# is the sum <r(2n 1) of all the divisors of 2n 1. (II) When 

 the second members are replaced by 4-3 A+2 (2m 1), then 



p + 2q = 3V(2m -- 1). 



(Ill) If A; is a prime 12X 1 and if 2n 1 is not divisible by k, while p, <? 

 are respectively the numbers of sets of distinct odd integral solutions not 

 all divisible by k of 



2z 2 + y* + z 2 = 4k K (2n - 1), z 2 + ?/ 2 + z 2 + * 2 = 4&"(2tt - 1), 



then p + q = A; lc ~ 1 X<r(2n - - 1). (IV) If M = a'tf -, where a, b, are 

 distinct odd primes, 4M is a sum of four odd squares without a common 



81 Nouv. Ann. Math., (2), 14, 1875, 90-91. 



62 Nouv. Corresp. Math., 4, 1878, 333, foot-note. 



63 Nouv. Ann. Math., (2), 17, 1878, 45. 

 M Giornale di Mat., 16, 1878, 152-167. 



