The number of representations of an integer as a sum of an odd number of squares
by
Shaun Cooper
Massey University - Albany

Let r_k(n) denote the number of solutions in integers of

x12+x22+ ... +xk2=n.

For example,

5=( +/- 2)2+( +/- 1)2 = ( +/- 1)2+( +/- 2)2,

and so r₂(5)=8, while r₂(3)=0. Geometrically, r_k(n) counts the number of lattice points on the k-dimensional sphere x₁²+x₂²+ ... +x_k²=n.

Formulas for r_k(n) for k=2,   4,   6 and 8 in terms of the divisors of n were essentially known to Jacobi. A general formula for r_k(n) for even k was given by Ramanujan.

The case when k is odd is more difficult. Some new results for odd k will be given. These were discovered by computer search, using Maple.

Date received: October 9, 2001