A sum of three nonunit squares of integers

TL;DR

This paper characterizes integers that can be expressed as the sum of three nonunit squares of integers under the assumption of the Generalized Riemann Hypothesis, extending to figurate numbers.

Contribution

It provides a complete classification of such integers under GRH and applies these results to figurate number sums for any k ≥ 3.

Findings

Identifies all integers as sums of three nonunit squares under GRH.

Extends results to sums of nonzero figurate numbers for any k ≥ 3.

Provides a theoretical framework assuming GRH.

Abstract

We say a positive integer is a sum of three nonunit squares if it is a sum of three squares of integers other than one. In this article, we find all integers which are sums of three nonunit squares assuming that the Generalized Riemann Hypothesis(GRH) holds. As applications, we find all integers, under the GRH only when $k=3$, which are sums of $k$ nonzero triangular numbers, sums of $k$ nonzero generalized pentagonal numbers, and sums of $k$ nonzero generalized octagonal numbers, respectively for any integer $k\ge 3$.