A sum of three nonzero triangular numbers

Abstract

Finding all integers which can be written as a sum of three nonzero squares of integers has been studied by a number of authors. This question is solved under the assumption of the Generalized Riemann Hypothesis (GRH), but still remains unsolved unconditionally. In this paper, we show that out of all integers that are sums of three squares, all but finitely many can be written as [Formula: see text] for some integers [Formula: see text]. Furthermore, we explicitly describe this finite set under the GRH. From this result, we also describe further generalizations for sums of nonzero polygonal numbers. Precisely, we find all integers, under the GRH only when [Formula: see text], which are sums of [Formula: see text] nonzero triangular (generalized pentagonal and generalized octagonal, respectively) numbers for any integer [Formula: see text].