On restricted sums of four squares and Zhi-Wei Sun's $x+24y$ conjecture
Authors
Hai-Liang Wu
Yue-Feng She
Abstract
In this paper, by using the arithmetic theory of ternary quadratic forms, we study some refinements on Lagrange's four-square theorem. For example, given positive integers $a,b$ satisfying some algebraic conditions and a positive integer $C\ge3$, we will show that for any sufficiently large integer $n$ with $\ord_2(n)\le C$, there exist non-negative integers $x,y,z,w$ such that
$$ \begin{cases}
x^2+y^2+z^2+w^2=n,
ax+by\in\mathcal{S},
\end{cases} $$
where $\mathcal{S}$ is the set of all squares over $\mathbb{Z}$. In particular, we obtain some progress on Zhi-Wei Sun's $x+24y$ conjecture.
