Factorials and Legendre's three-square theorem: II

Abstract

Let $\bar{S}$ denote the set of integers $n$ such that $n!$ cannot be written as a sum of three squares. Let $\bar{S}(n)$ denote $\bar{S}\cap [1,n]$. We establish an exact formula for $\bar{S}(2^k)$ and show that $\bar{S}(n)=1/8\ast n+\mathcal{O}(\sqrt{n})$. We also list the lengths of gaps appearing in $\bar{S}$. We make use of the software package Walnut to establish these results.

14 pages, 6 figures