Some variants of Lagrange's four squares theorem

TL;DR

This paper refines Lagrange's four squares theorem by proving that every nonnegative integer can be expressed as a sum of four squares with additional algebraic constraints on linear or quadratic combinations, using a new approach based on Euler's identity.

Contribution

It introduces novel refinements of Lagrange's theorem, establishing new representations with algebraic conditions, and advances the 1-3-5 conjecture through a fresh method involving Euler's four-square identity.

Findings

Every nonnegative integer can be written as a sum of four squares with linear sums being perfect powers.

New representations involve quadratic forms with sums being squares or cubes.

Progress on the 1-3-5 conjecture of Sun using a novel approach.

Abstract

Lagrange's four squares theorem is a classical theorem in number theory. Recently, Z.-W. Sun found that it can be further refined in various ways. In this paper we study some conjectures of Sun and obtain various refinements of Lagrange's theorem. We show that any nonnegative integer can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb Z)$ with $x+y+z+w$ (or $x+y+z+2w$, or $x+2y+3z+w$) a square (or a cube). Also, every $n=0,1,2,\ldots$ can be represented by $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb Z)$ with $x+y+3z$ (or $x+2y+3z$) a square (or a cube), and each $n=0,1,2,\ldots$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb Z)$ with $(10w+5x)^2+(12y+36z)^2$ (or $x^2y^2+9y^2z^2+9z^2x^2$) a square. We also provide an advance on the 1-3-5 conjecture of Sun. Our main results are proved by a new approach involving Euler's four-square identity