296 HISTORY OF THE THEORY OF NUMBERS. [CHAP, vm 



composition, determine the sign as before, and form the algebraic sum 

 B of the products. Then A = B, as shown by use of infinite series and 

 products. 



E. Catalan 71 noted that 



+ 2/ 4n = (- 



.2n+2 i ,,/2rH-2 



T. Pepin 72 gave a purely arithmetical proof that the number of repre- 

 sentations of m as a OB is 8{2 + ( l) m }X(m), where X(m) is the sum of 

 the odd divisors of m. The proof is like that by Jacobi 30 and Dirichlet. 42 

 Pepin 73 gave an exposition of this proof by Dirichlet and noted (p. 173) 

 that the theorem is a special case of one by Liouville; he proved (pp. 

 176-184) the theorems of Jacobi. 24 



M. Weill 74 noted that Jacobi deduced from the formula & 2 + k' 2 = 1 

 in elliptic functions the result that, if N is odd, the number of representa- 

 tions of 4N as a sum of 4 odd squares is double the number of representations 

 of N as a ffl, and gave a direct proof by means of the identity of Zehfuss. 46 

 By a similar identity, Weill proved that if TV" is any integer not divisible by 

 3, and if N and 3N admit only decompositions into four distinct squares 

 4= 0, the number of decompositions of 3N as a 30 is double the number of 

 those of N. 



G. Frattini 75 proved that the number of pairs of squares for which 

 z 2 Dy z = X (mod p) is %{p (Dip)}, where (Dip) is the quadratic 

 character of D with respect to the prune p. There is given an elegant 

 proof, due to Bianchi, of the existence of solutions if p > 3. If X is a 

 residue, take y = 0. If X is a non-residue, it is shown that, when a ranges 

 over the (p l)/2 residues, a \ is not always a residue and not always 

 a non-residue. For, if e = (p l)/2 and every root of x e = 1 satisfies 

 (x X) 6 = 1, it satisfies (x X) e x e = or 2, whereas the degree 

 is less than the number e of the roots. 



J. W. L. Glaisher 76 used the term partition (resolution) of N as a sum 

 of squares when we disregard the order in which the squares are placed 

 and the sins of the roots; composition when the order of the squares is 

 taken into account, but not the signs of the roots; representation when both 

 the order and the signs are attended to. For A 7 " odd, xW denotes the sum 

 of the square roots of the distinct squares appearing in the various partitions 

 of 2N into two squares, the sign + or being prefixed to each root accord- 

 ing as its numerical value is of the form 4n + 1 or 4n + 3. An equivalent 

 definition (p. 98) is that x(N) i g the sum of all the primary complex numbers 

 a + bi of norm N = a 2 + 6 2 . Two odd squares are said to be of the same 

 class if and only if both are of the form (Sn I) 2 or both of the form 



71 Nouv. Ann. Math., (3), 3, 1884, 347. 



72 Atti Accad. Pont. Nuovi Linoei, 37, 1883-4, 12-20. 

 Ibid., 38, 1884-5, 140-5. 



"Comptes Rendus Paris, 99, 1884, 859-861; Bull. Soc. Math. France, 13, 1884-5, 28-34. 

 76 Rendiconti Reale Accad. Lincei, (4), 1, 1885, 136-9. 

 78 Quar. Jour. Math., 20, 1885, 80-167. 



