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



J. W. L. Glaisher 129 wrote 4G(ri) for the excess of the number of repre- 

 sentations of n in the form (6r) 2 + (6s + I) 2 over the number of those in 

 the form (6r + 2) 2 + (6s + 3) 2 , provided n = 1 (mod 12), whence the 

 representations of n as a El are of one of those two types. If p, q are rela- 

 tively prime numbers 12k + 1, G(pr) = G(p)G(r). He evaluated G(a a }, a 

 being a prime. The number of representations of n as a El is E(ri), 

 where E(ri) is the excess of the number of divisors 4& + 1 of n over the 

 number of divisors 4& + 3. There are noted simple relations between 

 E(ri) and G(ri). It is shown (p. 195) by elliptic functions that the number 

 of representations of 4n + 1 as a sum of an even and an odd square is 

 4E(4n + 1) ; the number of representations of Sn + 2 as a sum of two odd 

 squares is 4E/(4n + 1). Hence if n = 1 (mod 4), n and 2n have the same 

 number of representations as EL Next, E(36n + 9) = E(4n + 1). The 

 number of compositions of a number as a sum of two squares, both of the 

 form (12n + I) 2 or both of the form (12n + 5) 2 , or one of each form, is 

 expressed in terms of functions E and G. Similarly for representations by 

 the forms at the beginning of this summary. Let (pp. 211-3) m be odd, 

 a even, b odd and not divisible by 3, c = 1, d = 5 (mod 12); then the 

 number of representations by 3a 2 + b 2 , 3a 2 + c 2 , 3a 2 + d 2 , 3m 2 + c 2 or 

 3m 2 + d 2 is expressed hi terms of G and the excess H(ri) of the number of 

 divisors = 1 (mod 3) of n over the number of divisors = 2 (mod 3). 



F. Goldscheider 13 discussed the sign of /, not determined by 

 Gauss. 44 



L. Gegenbauer 131 noted that Hermite's 124 formula is one of a set which 

 follows from a general formula for the sum of the values taken by an 

 arbitrary function f(y] when y ranges over all those divisors ^ V& x>f 

 k = 4n + 1 or 4n + 3. 



E. Lucas 132 gave two proofs by use of continued fractions that every 

 divisor of a sum of two relatively prime squares is a E] . 



K. Th. Vahlen 133 deduced from the theory of partitions the fact that 

 every odd integer is a El in E ways, if g 2 + u 2 and ( 0) 2 + u z are regarded 

 as different ways, while E is the excess of the number of factors 4m + 1 

 over the number of factors 4m + 3. He noted that this fact is equivalent 

 to the theorem of Jacobi 50 in view of a remark by Euler 24 (end). Since 

 every integer N is the product of an even power of 2 by an odd integer or 

 by the double of an odd integer, the number of sets of solutions ^ of 

 x 2 -f- 2/ 2 = N is E. He gave a summation formula for the number of primi- 

 tive representations as a El . 



From a representation a 2 + 6 2 + c 2 + d 2 of an odd prime p we obtain 

 a multiple of 32 representations by permuting a, , d or changing their 

 signs, except when two are zero, the factor being then 12-4. But there are 

 8<r(p) representations of p. Thus if p = a 2 + b 2 has N sets of solutions 



129 Proc. London Math. Soc., 21, 1889-90, 182-215. 



1JO Das Reziprozitatsgesetz der achten Potenzreste, Progr. Berlin, 1889, 26-29. 



i jl Sitzungsber. Akad. Wiss. Wien (Math.), 99, Ha, 1890, 387-403. 



132 ThSorie des nombres, 1891, 454-6. 



153 Jour, fur Math., 112, 1893, 25-32. 



