Factorials and Legendre's three-square theorem

TL;DR

This paper establishes a precise criterion, based on binary representation and automata, for when factorials can be expressed as sums of three squares, advancing understanding of number representations.

Contribution

It introduces a novel automaton-based condition linking binary digits of n to the sum-of-three-squares property of n!.

Findings

Characterization of n! as sum of three squares using binary automaton

Necessary and sufficient condition based on binary representation

New insights into factorials and sum-of-three-squares problem

Abstract

We provide a necessary and sufficient condition for $n!$ to be a sum of three squares. The condition is based on the binary representation of $n$ and can be expressed by the operation of an automaton.