Sums of four squares with a certain restriction

TL;DR

This paper proves conjectures by Sun that every positive integer can be expressed as a sum of four squares with a specific restriction on the variables, using advanced arithmetic theory of quadratic forms.

Contribution

It confirms Sun's conjectures on restricted sums of four squares through novel application of ternary quadratic form theory.

Findings

Every positive integer can be written as a sum of four squares with the restriction that x+3y is a perfect square.

For each positive integer n, there exist integers x,y,z,w such that n=x^2+y^2+z^2+w^2 and x+3y is a power of 4.

The conjectures are proved using arithmetic theory of ternary quadratic forms.

Abstract

In 2016, while studying restricted sums of integral squares, Sun posed the following conjecture: Every positive integer $n$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb{N}=\{0,1,\cdots\})$ with $x+3y$ a square. Meanwhile, he also conjectured that for each positive integer $n$ there exist integers $x,y,z,w$ such that $n=x^2+y^2+z^2+w^2$ and $x+3y\in\{4^k:k\in\mathbb{N}\}$. In this paper, we confirm these conjectures via some arithmetic theory of ternary quadratic forms.