Positive squares written as nontrivial sums of four squares

Zhi-Wei Sun

Published 2020-10-12Version 1

Let $m$ be any positive integer, and let $\lambda\in\{2,3\}$. We show that $m^2$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w$ nonnegative integers such that $x+2y+\lambda z$ is a positive square. We also pose some open conjectures; for example, we conjecture that any positive odd integer can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\{0,1,2,\ldots\}$ and $x+2y+3z\in\{2^a:\ a=1,2,3,\ldots\}$.