Refining Lagrange's four-square theorem

TL;DR

This paper refines Lagrange's four-square theorem by demonstrating specific algebraic and polynomial conditions under which any natural number can be expressed as a sum of four squares, introducing new constraints and conjectures.

Contribution

It introduces novel refinements of the four-square theorem involving polynomial and linear sum constraints, and proposes new conjectures for further exploration.

Findings

Any natural number can be expressed as a sum of four squares with linear sum conditions being a square or cube.

Certain polynomial conditions on the four squares can be satisfied in such representations.

Proposes conjectures, including the 1-3-5-Conjecture, about sums of four squares with specific linear constraints.

Abstract

Lagrange's four-square theorem asserts that any $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as the sum of four squares. This can be further refined in various ways. We show that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb Z$ such that $x+y+z$ (or $x+2y$, $x+y+2z$) is a square (or a cube). We also prove that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb N$ such that $P(x,y,z)$ is a square, whenever $P(x,y,z)$ is among the polynomials \begin{gather*} x,\ 2x,\ x-y,\ 2x-2y,\ a(x^2-y^2)\ (a=1,2,3),\ x^2-3y^2,\ 3x^2-2y^2, \\x^2+ky^2\ (k=2,3,5,6,8,12),\ (x+4y+4z)^2+(9x+3y+3z)^2, \\x^2y^2+y^2z^2+z^2x^2,\ x^4+8y^3z+8yz^3, x^4+16y^3z+64yz^3. \end{gather*} We also pose some conjectures for further research; for example, our 1-3-5-Conjecture states that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb N$ such that $x+3y+5z$ is a square.