On a Conjecture of Sun about sums of restricted squares

TL;DR

This paper advances understanding of sums of restricted squares, specifically addressing Sun's conjecture by generalizing classical results on sums of squares with restrictions on prime factorizations.

Contribution

It provides an ineffective generalization of Gauss and Legendre's results and an effective extension of Lagrange's four-square theorem for restricted prime factorizations.

Findings

Progress towards Sun's conjecture on sums of restricted squares

Generalization of classical sums of squares results

Effective version of Lagrange's four-square theorem

Abstract

In this paper, we investigate sums of four squares of integers whose prime factorizations are restricted, making progress towards a conjecture of Sun that states that two of the integers may be restricted to the forms $2^a3^b$ and $2^c5^d$. We obtain an ineffective generalization of results of Gauss and Legendre on sums of three squares and an effective generalization of Lagrange's four-square theorem.