General relations between sums of squares and sums of triangular numbers

Abstract

Let = ( 1, · · · , m) be a partition of k. Let r (n) denote the number of solutions in integers of 1x21 + · · · + mx2 m = n, and let t (n) denote the number of solutions in non negative integers of 1x1(x1 +1)/2+· · ·+ mxm(xm +1)/2 = n. We prove that if 1 k 7, then there is a constant c , depending only on , such that r (8n + k) = c t (n), for all integers n.