On restricted sums of four squares and Zhi-Wei Sun's $x+24y$ conjecture

TL;DR

This paper refines Lagrange's four-square theorem using ternary quadratic forms, establishing conditions for representing large integers with additional algebraic constraints, and advances understanding of Zhi-Wei Sun's $x+24y$ conjecture.

Contribution

It introduces new conditions for representing integers as sums of four squares with algebraic constraints, improving upon classical results and making progress on Sun's conjecture.

Findings

Established conditions for representing large integers with four squares and algebraic constraints.

Proved that for sufficiently large integers, certain linear forms involving squares are representable.

Made progress towards Zhi-Wei Sun's $x+24y$ conjecture.

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.