强24猜想

中文摘要 AI 辅助

拉格朗日四平方定理的限制形式，其中表示变量需要满足额外的算术条件，已从多个角度进行了研究。在Zhi-Wei Sun提出的改进中，有一个猜想：每个非负整数都有一个表示\\[ N=x^2+y^2+z^2+w^2,\qquad x,y,z,w\in\mathbb Z_{\ge0},\\]使得\\[ x\\ \text{和}\\ x+24y\\ \text{是完全平方数}。\\]我们证明了这个猜想对所有足够大的整数成立。证明从两个平方条件的初等参数化开始，并结合半线性二平方筛法与泊松求和、有限域相消和切换方法。

英文摘要

Restricted forms of Lagrange's four-square theorem, in which the representing variables are required to satisfy additional arithmetic conditions, have been studied from several points of view. Among the refinements proposed by Zhi-Wei Sun is the conjecture that every nonnegative integer has a representation \[ N=x^2+y^2+z^2+w^2,\qquad x,y,z,w\in\mathbb Z_{\ge0}, \] for which \[ x\ \text{and}\ x+24y\ \text{are perfect squares}. \] We prove this conjecture for all sufficiently large integers. The proof starts from an elementary parametrization of the two square conditions and combines a semi-linear two-squares sieve with Poisson summation, finite-field cancellation, and switching.

发表机构

• Georg-August-Universität Göttingen（哥廷根大学）
• Aba Teachers University（阿坝师范学院）

机构由 AI 辅助整理，请以论文原文为准。