288 HISTORY OF THE THEORY OF NUMBERS. [CHAP, vili 



m = a a b 3 , where a, b, are distinct primes, the number of proper 

 representations is 



8m(l + l/a)(l + 1/6) . 



P. L. Tchebychef 33 proved that x 2 Ay 2 B = (mod p) is solvable 

 if A is not divisible by the prime p. Proof is needed only when p > 2 and 

 Ay 2 + B is never divisible by p, whence 



(Ay 2 + )(p-i>/ 2 + 1 = (mod p). 



This congruence of degree p 1 is not satisfied by all the values 0,1, , 

 p 1 of y, so that for one of them Ay 2 + B is a quadratic residue of p. 



F. Pollock 34 noted that if any odd square 16n 2 db 8n + 1 is increased by 

 3 the sum is 3(4n 2 4n + 1) + (4n 2 =F 4n + 1), and hence is the sum of 

 four odd squares. By adding also 8, the new sum is divisible into four odd 

 squares, with a like result for each addition of 8. He stated that every 

 number Sk + 4 is reached in this way. Since every number 8k + 4 is thus 

 a 00, Bachet's theorem is said to follow. 



C. Hermite 35 showed that, if A is odd or the double of an odd number, 



(12) a 2 + (3 2 + 1 = (mod A) 



has integral solutions. First, let A = e (mod 4), = =t 1. The arith- 

 metical progression with the general term 4Az + 2eA 1 contains by 

 Dirichlet's theorem an infinitude of primes, each = 1 (mod 4) and hence 

 the sum of two squares a 2 + /3 2 - Next, let A = 2 (mod 4) ; we employ 

 similarly the progression 2Az + A 1. 



For integral solutions a, of (12), the definite form 



/ = (Ax + az + Pu} 2 + (Ay - (3z + cm) 2 + z 2 + u 2 



has as the numerical value of the invariant A the value A* (being the product 

 of the square of the determinant A 2 of the four linear functions by the value 

 1 of A for the sum of 4 squares) and hence its minimum for integral values 

 of the variables x, , u is < (-f) 3/2 A 1/4 < 2 A. Since / represents only 

 multiples of A, the minimum is A itself. Thus A can be represented by / 

 and hence is a sum of four squares. 



Hermite 36 repeated the preceding proof and gave the following. The 

 form 



i-/ = A(x 2 + y 2 ) + 2<x(zx + yu) + 2(3(xu -zy) + ^- (a 2 + p 2 + l)(z 2 + u 2 ) 



has integral coefficients, and A = 1. Hence it is equivalent to 



X 2 + Y 2 + Z 2 + U\ 



the single reduced definite quaternary form with A = 1. Hence in the 

 four linear functions X, , U of x, - , u, the sum of the squares of the 

 coefficients of x or of y equals A. 



"Theorie der Congruenzen, in Russian, 1849; in German, 1889, 207-9. 



M Proc. Roy. Soc. London, 6, 1851, 132-3. 



S5 Comptes Rendus Paris, 37, 1853, 133-4; Oeuvres, I, 288-9. 



88 Jour, fur Math., 47, 1854, 343-5, 364-8; Oeuvres, I, 234-7, 258-263. 



