Restricted sums of four squares

TL;DR

This paper refines Lagrange's four-square theorem by establishing new restrictions involving powers of two and other algebraic conditions, demonstrating that all positive integers can be represented as sums of four squares under these constraints.

Contribution

The paper introduces novel restrictions on four-square representations involving powers of two and other algebraic conditions, expanding classical results with new structured representations.

Findings

Every positive integer can be expressed as a sum of four squares with specific algebraic restrictions.

New representations involve conditions like sums or differences being powers of four or cubes.

The paper proposes open conjectures related to sums involving squares and powers of integers.

Abstract

We refine Lagrange's four-square theorem in new ways by imposing some restrictions involving powers of two (including $1$). For example, we show that each $n=1,2,3,\ldots$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb N=\{0,1,2,\ldots\})$ with $|x+y-z|\in\{4^k:\ k\in\mathbb N\}$ (or $|2x-y|\in\{4^k:\ k\in\mathbb N\}$, or $x+y-z\in\{\pm 8^k:\ k\in\mathbb N\}\cup\{0\}\subseteq\{t^3:\ t\in\mathbb Z\}$), and that we can write any positive integer as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb Z)$ with $x+y+2z$ (or $x+2y+2z$) a power of four. We also prove that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+2w^2$ $(x,y,z,w\in\mathbb Z)$ with $x+y+z+w$ a square (or a cube). In addition, we pose some open conjectures for further research; for example, we conjecture that any integer $n>1$ can be written as $a^2+b^2+3^c+5^d$ with $a,b,c,d\in\mathbb N$.