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

TL;DR

This work advances constrained representations in additive number theory by proving that, under precise coprimality and unit-group conditions on (a,b), one can represent sufficiently large integers n as a sum of four squares with the linear form ax+by forced to be a square. The authors employ the arithmetic theory of ternary quadratic forms, together with genus and spinor-genus analysis, to establish a local-global framework that ensures the desired representation for all large n with controlled 2-adic size. A corollary yields progress toward Zhi-Wei Sun’s x+24y conjecture, and the results illustrate a systematic method for enforcing square-restrictions on linear forms within four-square decompositions. The approach integrates detailed p-adic lattice analysis with adelic methods to produce constructive representations and illuminates the necessity of the ord_2(n) condition in general.

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.

Theorems & Definitions (10)

• Conjecture 1.1: Z.-W. Sun
• Theorem 1.1
• Remark 1.1
• Corollary 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6