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



Legendre 40 stated that every divisor of a sum of two relatively prime 

 squares is a sum of two relatively prime squares. P. Volpicelli 41 noted 

 that the latter need not be relatively prime since d = 2197 = 39 2 -f 26 2 

 is a divisor of 13d = 119 2 + 120 2 [but d also equals 9 2 + 46 2 ]. 



P. Barlow 42 stated that a number 4n + 1 is a prime if a El in one way 

 only. [He should have said relatively prime squares; 45 = 36 + 9 is a EJ 

 in a single way. For Euler's proofs of the correct theorem see Ch. XIV 

 of Vol. I]. 



A. Cauchy 43 obtained (1) by taking the norm of the product of two 

 complex numbers. 



C. F. Gauss 44 stated that, if a prime p = 4k + 1 is expressed in the form 

 e 2 + / 2 , e odd, / even, then e and db / equal the minimum residues (i. e., 

 between - p/2 and + p/2) modulo p of Jr/(*I) and |r 2 , respectively, where 



r = (k + l)(& + 2) ... (2k). 



The residue of db e is positive or negative according as the positive value 

 of e is of the form 4m + 1 or 4m + 3. But there is given no general rule 

 as to the sign of / (cf. Golds cheider 130 ). 



Gauss 45 noted that the number of sets of integers x, y for which 

 x z + y 2 ^ A is 



4? 2 + 1+ 4[VZ] + 8 Z [>5 -j 2 ] 



j=g+i 



= 1 + 4{[A] - [A/3] + [A/5] - [A/7] +}, 



where # = [VZ/2], r = q + [^A], and p] denotes the greatest integer ^ t. 

 Denote by /(A) the number of representations of A by x 2 + i/ 2 , which i s 

 8 if A is a prime 4n + 1, while for A = 2 li Sa a b^ -> (as in Gauss 37 ) 



/(A) = 4(a + l)(/3 + !) 



or 0, according as S is a square or not. The mean of /(A) is IT. Set 

 /'( m ) = f( m ) + /(3m) ; the mean of f(m) is 4ir/3. Set 



/"(m) =/'(5m) -f(m); 



the mean of /"(w) is 167T/15. Proceeding, we approach the mean 4 and 

 find that 



4 = TT . -J . f f H (to infinity), 



the denominators being the successive odd primes p, and the numerators 

 being pl. 



Th6orie des nombres, 1798, 190; ed. 2, 1808, 175; ed. 3, 1830, I, 203 (Maser, I, 204). 



Annali di Sc. Mat. Fis., 4, 1853, 296. 



42 Theory of Numbers, London, 1811, p. 205. 



Cours d'analyse de 1'ecole polyt., 1, 1821, 181. 



Gott. gelehrte Anz., 1, 1825; Comm. soc. sc. Gott. recent., 6, 1828; Werke, II, 1863, 168, 



90-1. Cf. Bachmann, 96 Kreisteilung, Ch. X. 

 Posth. MS., Werke, II, 1863, 269-275, 292; Gauss-Maser, Hohere Arith., 1889, 656-661. 



Cf. Eisenstein, 66 Hermite. 117 - m 



