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



De Rocquigny 72 obtained a solution of 02 SI = S] by use of 

 (a 2 + Xb 2 )(ai + X&i) = (ai + X&&0 2 + X(a&! - ai&) 2 , X = c 2 + d 2 . 



Catalan 73 took the fourth variables zero in Euler's (1) of Ch. VIII 

 and got 



+ zzi) 2 + (xy l - yxj 2 + (yz l - zyj 2 + (zx l - 



Taking x : Xi = y : y\ [Young 27 ], we get P = SI ; but the condition is not 

 necessary in view of (9 + 4 + 1)(1 + 1 + 1) = 25 + 16 + 1. 



K. Th. Vahlen 74 deduced (1) from the theory of partitions. The identity 



a 2 + 2/3 2 + 37 2 + 65 2 = a 2 + (/3 + 7 + 5) 2 + (- /3 + 7 + 5) 2 + (7 - 25) 2 



and Jacobi's 23 final result shows that every number is a HI. 



Catalan 75 proved that if p is not a GO, then p 2 is a SI. For, if 



p = a 2 + 6 2 + c 2 , p 2 = (a 2 + 6 2 - c 2 ) 2 + (2ac) 2 + (26c) 2 . 

 If p = a 2 + 6 2 + c 2 + d 2 , then 



Catalan 76 noted that every odd square > 1 is a sum of 2 or 3 squares. 



P. Bachmann 77 considered the number A of decompositions of s into 

 three distinct squares a 2 + i + z where one (or three) of a, i, 2 is of 

 the form 12k 1 and the others are of the form 12k 7; the number A' 

 of decompositions into three distinct squares for which the reverse is true; 

 the number B of decompositions s = a 2 + 2i in which a, i are distinct 

 and a is of the form 12k 1 ; and the number 5' of such decompositions 

 in which a = 12& 7. He proved that 2A + B = 2A f + B f + D, where 

 D = or {( l) l '(2i + 1) j}/3, according as s is not or is of the form 

 3(2i + I) 2 , and j is the absolutely least residue modulo 3 of ( l)*'(2i + 1). 



Bachmann 78 gave an exposition of the theory of G3 . 



J. F. d'Avillez 79 applied Catalan's 43 formula to express the squares of 

 1, 3, 6, 11, 17, 25, 34, 45, as SI. 



We may express 1521 as a S] in 7 ways. 80 Many identities giving equal 

 sums of three squares have been noted. 81 



M. A. Gruber 82 tabulated solutions of 3 2n = S] forn ^ 6. 



R. D. von Sterneck 83 gave an elementary proof of (1). 



72 Mathesis, (2), 2, 1892, 136. 



73 Ibid., (2), 3, 1893, 105-6. 



74 Jour, fur Math., 112, 1893, 23. 



78 M6m. Acad. Roy. Belgique, 52, 1893-4, 21. 



76 Mathesis, (2), 4, 1894, 27, 52-53. 



77 Die Analytische Zahlentheorie, 1894, 37-9. 



78 Arith. der Quadrat. Formen, 1898, 139-162, 600; Niedere Zahlentheorie, 2, 1910, 320-323. 



79 Jornal de Sc. Math. Phys. e Nat., (2), 5, 1897-8, 90-2. 



80 Amer. Math. Monthly, 5, 1898, 214. 



81 Ibid., 6, 1899, 17-20. 

 S2 Ibid., 8, 1901,49-50. 



83 Sitzungsber. Akad. Wiss. Wien (Math.), 109, Ha, 1900, 28-43. 



