CHAP. IX] SUM OF FlVE OR MORE SQUARES. 313 



Minkowski, 28 then a student of 18 years of age at the University of Konigs- 

 berg. 



Ch. Berdelle' 33 proved that any multiple of 8 is a sum of 8 odd squares. 

 From 



n = a 2 + & 2 + c 2 + d 2 , 8a 2 = 4a 2 - 4a + 4a 2 + 4a, 

 8 + 8n is the sum of the squares of 

 2a + 1, 2a- 1, 26 + 1, 26 - 1, 2c + 1, 2c - 1, 2d +1, 2d - 1. 



If k of the integers a, 6, c, d are zero, 2k of the 8 squares are unity. 



J. W. L. Glaisher 34 noted that, if <r(ri) is the sum of the divisors of n, 

 the number of representations of n as a sum of five squares is 



10{<r(n) + 2<r(n - 4) + 2a(n - 16) + } if n = 1 (mod 8), 



but twice that expression if n = 3 (mod 4). 



L. Gegenbauer 35 proved that the number of representations of an odd 

 number n as a sum of eight squares equals 16M, where M is the number of 

 divisors of the various g.c.d.'s of n with all triples chosen from 1, , n. 

 Also M is the sum of the products of the number of divisors of every factor 

 of n by the number of those triples whose elements do not exceed the com- 

 plementary divisor and form a system relatively prune to it. There are 

 three further theorems on sums of 8 squares, five on sums of 12 squares 

 and two on sums of 6 and 10 squares each. The number of all [or proper] 

 representations of an odd number n as a sum of three squares and double 

 a square is 2 {4 (2/w) }/x, where the symbol is Jacobi's and /* is the num- 

 ber of all [or proper] representations x 2 2y z , y ^ 0, 2x > 3y, of the 

 various g.c.d.'s of n and the numbers ^ n. There is a similar theorem 

 on a sum of five squares and double a square. 



G. B. Mathews 36 noted that the number of sets of solutions of 



x\ + - - + xl = n 



is the coefficient c of q n in the expansion of 



l+q 1-q 2 1+q* 



LfK I I / rt / V>^ ,1 sft<y i *v fj ^ "* -* 



" IX .'/ *-'/ **'/ 7 * v - "" * * * 



1 q 1+5 2 1 q 3 

 By logarithmic differentiation, 



ide 



e dq & Xn}qn l > m = 2 



summed for all odd divisors IJL of n. For 34 n = 2 a m, \J/(ri) = 2 a+ V(m). By 

 the logarithmic differentiation of 6 k 1 + c\q + c z q 2 + and comparison 



33 Bull. Soc. Math, de France, 17, 1888-9, 102, 205. Cf. Catalan. 16 



34 Messenger Math., 21, 1891-2, 129-130. 



35 Sitzungsber. Akad. Wiss. Wien (Math.), 103, Ha, 1894, 122-5. 



36 Proc. London Math. Soc., 27, 1895-6, 55-60. 



