Sums of four squares with a certain restriction

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.