Factorials and Legendre's three-square theorem: II

TL;DR

This paper investigates integers related to factorials that cannot be expressed as sums of three squares, providing exact formulas, asymptotic estimates, and analyzing the distribution of such integers using computational tools.

Contribution

It establishes an exact formula for the set of integers where factorials are not sums of three squares and analyzes their distribution with computational methods.

Findings

Exact formula for $ar{S}(2^k)$

Asymptotic estimate $ar{S}(n) = 1/8 * n + O(\sqrt{n})$

List of gap lengths in $ar{S}$

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 $\bar{S}(n) = 1/8*n + \mathcal{O}(\sqrt{n})$. We also list the lengths of gaps appearing in $\bar{S}$. We make use of the software package Walnut to establish these results.