CHAP. VIII] SUM OF FOUR SQUARES. 289 



For M an odd integer, the Hermitian form 



MVV + (a + tf)VU + (a- 0i)VU + (a 2 + /3 2 + 1)UU, 



with complex integral coefficients, has for the invariant A the value 1, 

 and hence is equivalent to vv + uu, the single reduced form with A = 1. 

 Let the latter be transformed into the former by 



v = aV + bU, u = cV + dU, ad - be = 1, 



a, - , d being complex integers. Then M = ad + cc, where a and c are 

 relatively prime. Thus any odd integer is the sum of four squares such 

 that the sum of two of the squares is prime to the sum of the two remaining 

 squares. 37 



By considering the proper and improper representations of M by 

 vv + uu, he obtained Jacobi's formula 811 (pi + 1) for the number of repre- 

 sentations as a sum of 4 squares of M = lip,, when M is not divisible by 

 the square of a prime. 



F. Pollock 38 proved Cauchy's theorem (1813) that any odd number 

 2p + 1 is a sum of four squares the algebraic sum of whose roots is any 

 assigned odd number from 1 to the maximum. For, p is a sum of three or 

 fewer triangular numbers. If p = (q 2 + g)/2, then whether q = 2n or 

 2n 1, we have 2p + 1 = 4n 2 2n + 1, which is the sum of the squares 

 of n, - n, =F n, d= (n 1). If p = (g 2 + q)/2 + (r 2 + r)/2, then p is of 

 the form a 2 + a + & 2 , and 2p + 1 is the sum of the squares of a -f- 1, a, 

 6, b. If p is the sum of three triangular numbers, 



p = a? + a + 6 2 + |(m 2 + w), 

 2p + 1 = 2(a 2 + a + 6 2 ) + 4n 2 2rc + 1, 



the latter being the sum of the squares of b =F n, 6 =F n, a n, 

 a n + 1. In every case the algebraic sum of the four roots is unity. 



A. Genocchi 39 " recalled ' (without reference) formulas (7) and (8) 

 and noted that the second implies that the number of representations of 

 4n as a 30 is a(ri) when n is odd, and that the first implies 



Ni + 2N 2 + W 3 + SN* = 41*! + D 2 - A 



where DI is the sum of the odd divisors of n, Z> 2 (or Z) 4 ) the sum of the even 

 divisors d of n with njd odd (or even), while NI, , A^ is the number of 

 solutions of x\ + + x\ = n with 3, 2, 1, unknowns zero. For another 

 similar formula see Cesaro 30 of Ch. IX. 



A. Desboves 40 stated empirically that the double of any odd integer is 

 a sum of two primes 4n + 1. Such a prime is a 12. Hence every integer 

 is a SI . 



87 E. Picard, the editor of Hermite's Oeuvres, 1, p. 259, noted that when a and c are relatively 

 prime, aa and cc are not necessarily so; but that the theorem in the text is probably true. 



38 Phil. Trans. Roy. Soc. London, 144, 1854, 311-9. 



39 Nouv. Ann. Math., 13, 1854, 169. 



40 Nouv. Ann. Math., 14, 1855, 293-5. 

 20 



