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arXiv · 1805.03038

A sum of squares not divisible by a prime

Kyoungmin Kim·Byeong-Kweon Oh
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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 $79$.

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Kyoungmin Kim, Byeong-Kweon Oh. 2018-05-08. A sum of squares not divisible by a prime. https://arxiv.org/abs/1805.03038

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