Sums of four squares with a certain restriction

Yue-Feng She; Hai-Liang Wu

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arXiv · 2010.02067

Sums of four squares with a certain restriction

Yue-Feng She·Hai-Liang Wu

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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.

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Yue-Feng She, Hai-Liang Wu. 2020-10-05. Sums of four squares with a certain restriction. https://arxiv.org/abs/2010.02067

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