CHAP. VI] SUM OF TWO SQUARES. 249 



conversely. In particular, all rational automorphs of x\ + x\ are derived 

 by taking X and X i2 to be relatively prime integers. To show (p. 384) 

 that every prime p = 4r + 1 is a El , use a solution of co 2 + 1 = (mod p) 

 and set 1 = o> 2 , where 2 is any integer not divisible by p. We can choose 

 relatively prime integers p , p 2 i such that rp and rp 2 i are numerically 

 < p/2 and congruent modulo p to 1 and 2 respectively. Take Pi 2 = - - p 2 i. 

 Then r 2 (po + Pi 2 ) is < ^p 2 and is divisible by p. Hence p* + p? 2 = pt, where 

 t < p/2. Determine </> and <i 2 numerically < t/2 and congruent modulo 

 t to po and p 12 respectively. Then $1 + <? 2 = it', where ' ^ t/2. Then 



where pi, p| 2 are relatively prime. Hence p' + p' 12 = pk, k = t'/r'* ^ </2. 

 Repeating this process, we finally get X = p ( s) , Xi 2 = PU, such that 

 Xo + Xfa = P, and 

 (6) X i Xi 2 2 = 0, Xi 2 i + X 2 = (mod p). 



Similarly we can find a complex integer with relatively prime coordinates 

 Xo, Xi2, whose norm is any power p y of p and which satisfies (6) modulo p y . 

 If m = p y q s , where p, g, are primes = 1 (mod 4), or if m is the 

 double of such a product, apply the preceding discussion for each p y and 

 take the product of the resulting complex integers. By using all sets of 

 solutions of 1 -f = (mod p y ), we get every proper representation of 

 m as a 03 and each once and but once. 



C. Hermite 124 proved by use of elliptic functions that, if M = 4n + 1, 



[M m z ~\ 

 --^ J, 



summed for m = 1, 3, 5, , where /(w) is the number of representations 

 of n as a 02 . The asymptotic value of S is |Af IT. 



A. Berger 125 gave an elementary proof of the theorem that, if n is a 

 positive odd integer, the number of all sets of solutions of x z + y 2 = n 

 is 42 ( l)( a - 1 )/ 2 , where 6 ranges over all positive divisors of n. While 

 Dirichlet's proof was by transcendental analysis, Berger uses only the 

 known number (Dirichlet 88 ) of relatively prime sets of solutions. 



Berger 126 proved that if n is a positive integer the number of sets of 

 integers x, y for which z 2 + y 2 = n is 42 sin d-rr/2 (Berger 125 ). 



C. Hermite 127 proved Gauss' 45 formula for the number of sets of integers 

 x, y for which x 2 + y 2 =i A. 



E. Catalan 128 noted that, if x 2 + y z + z 2 is a square, 



If B 2 - AC = - m 2 , (Ca - Ac) 2 - 4(Bc - Cb)(Ab - Ba) = 



Jour, fiir Math., 99, 1886, 324-8; Oeuvres, IV, 209-214. Cf. Gegenbauer. 131 



125 Acta Math., 9, 1886-7, 301-7. 



126 Ofversigt af Kongl. Vetenskaps-Akad. Forhandl., 44, 1887, 153-8. 



127 Amer. Jour. Math., 9, 1887, 381-8; Oeuvres, IV, 241-250. 

 8 Mathesis, 7, 1887, 120, 144. 



