A sum of squares not divisible by a prime

TL;DR

This paper investigates the minimal number of squares not divisible by a prime needed to represent all positive integers, establishing exact values for various primes and identifying a unique exception.

Contribution

It determines the exact values of S(p) for primes 2, 3, 5, and all primes greater than 5, providing new bounds and a specific exception.

Findings

S(2)=10

S(3)=6

S(5)=5','S(p)=4 for p>5

Abstract

Let $p$ be a prime. We define $S(p)$ the smallest number $k$ such that every positive integer is a sum of at most $k$ squares of integers that are not divisible by $p$. In this article, we prove that $S(2)=10$, $S(3)=6$, $S(5)=5$, and $S(p)=4$ for any prime $p$ greater than $5$. In particular, it is proved that every positive integer is a sum of at most four squares not divisible by $5$, except the unique positive integer $79$.