248 HISTORY OF THE THEORY OF NUMBERS. [CHAP. VI 



an even number of primes exceeds the number of representations of the 

 remaining divisors by the excess of the number of those divisors, with 

 complementary square divisor, of the form 4s + 1 over the number of such 

 divisors of the form 4s 1. 



T. Pepin 120 quoted Dirichlet's 52 theorem that the number of representa- 

 tions of an odd number n by x 2 + y 2 is 4p, where 



P = 



is a sum of Legendre-Jacobi symbols. It follows readily that the number 

 of representations of 2n is 4p and the number of decompositions is p. Since 

 p is the excess of the number of divisors 41 + 1 over the number of divisors 

 41 + 3, we have Jacobi's 50 theorem that the number of decompositions of 2n 

 is that excess. Likewise, 2 a n = x 2 + y 2 has 4p solutions. 



S. Re*alis 121 noted that if p is a prime or a product of primes of the 

 form 4q -f 1, all integral solutions of x 2 + y 2 = p are found from the 

 identity 



by giving to a and b such integral values that the second member takes the 

 value p. Thus the problem reduces to that of expressing q as a sum of two 

 triangular numbers. If p is odd or the double of an odd number and if 

 p = x 2 + y 2 ) where x and y are relatively prime, then 



(m 2 m n 2 T n\ 

 ~~2~ ~~2~~)' 



J. W. Bock 122 employed the n(2n 1) pairs formed by two of I 2 , 2 2 , > 

 (2n) 2 . From any pair xl, yl, whose sum is not divisible by the prime 

 p = 4n + 1, we obtain 2n incongruent sums v 2 xl + v 2 yl, v = 1, -, 2n. 

 If xl -f- yl is not congruent to one of these sums, nor to zero, it leads to 2n 

 new sums v 2 xl + v 2 yl; etc. But 2n does not divide n(2n 1). Hence there 

 exists a sum s = A 2 + B 2 divisible by p, < A < %p, < B < f p. In 

 the attempt to prove that, if s is divisible by a prime q = a 2 + b 2 , the 

 quotient is a sum of two squares, the quotient is taken to be c 2 + d 2 , c and 

 d not being assumed integral. By (1), q(c 2 + d 2 ) is of the form x 2 -f- y 2 . 

 From s = x 2 + y 2 , it is concluded erroneously that A = x or y, B = y or x. 



R. Lipschitz 123 noted that all real substitutions of determinant unity 

 for which xl + xl = yl + yl (i- e., automorphs) are given by multiplying 



(X + i\n}(xi + ix 2 ) = (X ^Xi 2 )(2/i + iyz) 

 by Xo iX 12 and equating the real terms and the imaginary terms, and 



120 Atti Accad. Nuovi Lincei, 38, 1884-5, 166. 



121 Nouv. Ann. Math., (3), 4, 1885, 367-9; Oeuvres de Fermat, IV, 218-220. 



122 Mitt. Math. Gesell. Hamburg, 1, 1885, 101-4. 



113 Untersuchungen iiber die Summen von Quadraten, Bonn, 1886, 147 pp. French transl. 

 by J. Molk, Jour, de Math., (4), 2, 1886, 373-439. Summary in Bull, des Sc. Math. 

 Astr., (2), 10, I, 1886, 163-183. 



