CHAP. VI] SUM OF TWO SQUARES. 247 



E. Catalan 114 expressed s = z 4n+2 + y* n+z as the sum of the squares of 

 two polynomials, and s 2 as such a sum in two ways (p. 51). By use (p. 63) 

 of (x iy)(x 2 iy 2 ) (x^' 1 ly 2 "" 1 ) = P + iQ, we get 2"" 1 decomposi- 

 tions of (x 2 + ?/ 2 ) O 4 -f ?/ 4 ) 2n + 2/ 2n ) as a El. 



Catalan 115 noted that, if a + b = El, and n = 2 P , 



a n-l 



C. Hermite 116 stated that if f(ri) is the number of solutions of x 2 + y 2 = n, 



+ 2) 



where #iO) = [a; + |] - [x] = [2z] - 2[>] is the function used by 

 Gauss. 



Hermite 117 proved by use of expansions of elliptic functions 



s =/(!) +/(2) + - - +/(O = 4S(- l^-^CC/a], 



* = /(2) + /(10) + - - - + /(8C + 2) = 4S(- 1)- 1 [(2(7 + c)/(2c - 1) J 



summed for a = 1, 3, 5, ; c = 1, 2, 3, . He stated that 



-"sin'-, 



where n = [( VgC + 1 + l)/4]. Also, for w = [( V4C + 1 + l)/2], 

 t = 8 | I - I - [ ~3 12 J+ [ ~ 5 2 3 J ~ ' 



J. c,^2 _. 



He proved Gauss' 45 result for s; also, J. Liouville's 118 result 



+ 2 - a 2 + 1)]. 



L. Gegenbauer 119 concluded from a general theorem on quadratic forms 

 that the number of ways any number r which is odd or the double of an odd 

 number can be represented as a sum of two squares is the quadruple 

 of the number of decompositions into two relatively prime factors of 

 those divisors of r which have only prime factors of the form 4s + 1 and 

 a square as complementary factor. The number of representations by 

 x 2 -f y 2 of those divisors of r whose complementary divisor is a product of 



114 Atti Accad. Pont. Nuovi Lincei, 37, 1883-4, 80. 



115 Mathesis, 4, 1884, 70. 



116 Amer. Jour. Math., 6, 1884, 173-4. 



117 Bull. Ac. Sc. St. Petersbourg, 29, 1884, 343-7 (Oeuvres, IV, 159-163); reprinted, Acta 



Math., 5, 1884-5, 320. 



118 Jour, de Math., (2), 5, 1860, 287-8. 



119 Sitzungsber. Akad. Wiss. Wien (Math.), 90, II, 1884, 438. 



