Representing sets with sums of triangular numbers

Abstract

We investigate here sums of triangular numbers f (x) := ∑ i b iT xi where T n is the nth triangular number. We show that for a set of positive integers S, there is a finite subset S 0 such that f represents S if and only if f represents S 0. However, computationally determining S 0 is ineffective for many choices of S. We give an explicit and efficient algorithm to determine the set S 0 under certain generalized Riemann hypotheses, and implement the algorithm to determine S 0 when S is the set of all odd integers. © The Author 2009. Published by Oxford University Press. All rights reserved.