Sums of squares of integers from residue classes

TL;DR

This paper investigates the minimal number of squares needed to represent large integers as sums of squares from specific residue classes, establishing conditions for universality and calculating minimal sums.

Contribution

It characterizes when residue classes are s-square universal and determines the minimal number of squares needed in these cases.

Findings

Residue class $ ext{A}_{d,m}$ is s-square universal iff $d ot ot ext{equiv} ext{to} \pm 1 ext{mod} ext{m}$.

The minimal number $ ext{SU}( ext{A}_{d,m})$ is explicitly determined for universal classes.

The paper provides a complete characterization of s-square universality for residue classes.

Abstract

A subset $\mathcal{A}\subseteq\mathbb{Z}$ is called $s$-almost square universal if every sufficiently large positive integer can be written as a sum of at most $s$ squares of integers from $\mathcal{A}$. In this article, we study the minimal number $\mathrm{ASU}(\mathcal{A}_{d,m})$ with this property, where $\mathcal{A}_{d,m}$ denotes the residue class of $d$ modulo $m$, with $m\in\mathbb{N}$ and $d\in\mathbb{Z}$. We further prove that $\mathcal{A}_{d,m}$ is $s$-square universal for some $s\in\mathbb{N}$ if and only if $d \equiv \pm 1 \pmod{m}$, and determine the minimal such number $\mathrm{SU}(\mathcal{A}_{d,m})$ in these cases.