A sum of squares not divisible by a prime

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.