# Sums of integers and sums of their squares

## TL;DR

This paper explores the possible sums of integers in representations of a number as a sum of squares, extending known results for up to 11 terms using Mordell's theory, and generalizes to linear forms.

## Contribution

It characterizes the set of possible sums for all representations with up to 11 squares and generalizes the problem to linear forms, building on Mordell's theory.

## Key findings

- Characterized $ ext{S}_m(n)$ for all $m \,\leq\, 11$
- Extended results to linear forms in integers
- Reinterpreted and generalized earlier results by Sun et al.

## Abstract

Suppose a positive integer $n$ is written as a sum of squares of $m$ integers. What can one say about the value $T$ of the sum of these $m$ integers itself? Which $T$ can be obtained if one considers all possible representations of $n$ as a sum of squares of $m$ integers? Denoting this set of all possible $T$ by $\mathscr{S}_m(n)$, Goldmakher and Pollack have given a simple characterization of $\mathscr{S}_4(n)$ using elementary arguments. Their result can be reinterpreted in terms of Mordell's theory of representations of binary integral quadratic forms as sums of squares of integral linear forms. Based on this approach, we characterize $\mathscr{S}_m(n)$ for all $m\leq 11$ and provide a few partial results for arbitrary $m$. We also show how Mordell's results can be used to study variations of the original problem where the sum of the integers is replaced by a linear form in these integers. In this way, we recover and generalize earlier results by Z.W. Sun et. al..