On Graham partitions twisted by the Legendre symbol

Abstract

We investigate when there is a partition of a positive integer n, n = f(lambda(1)) + f(lambda(2))+& ctdot;+f(lambda(& ell;)),satisfying that 1 = chi(p)(lambda(1))/lambda(1 )+ chi(p)(lambda(2))/lambda(2)+& ctdot; + chi(p)(lambda(& ell;))/lambda(& ell;,)where chi(p) is the Legendre symbol modulo prime p and f(k) = k or the k th m -gonal number with m = 3, 4, or 5.