Numbers represented by restricted sums of four squares

TL;DR

This paper investigates specific representations of nonnegative integers as sums of four squares with additional algebraic restrictions, utilizing quaternion arithmetic in Lipschitz integers.

Contribution

It introduces new conditions on sums of four squares and employs quaternion algebra to prove these restricted representations.

Findings

Every nonnegative integer can be expressed as a sum of four squares with a linear combination being a square or cube.

Uses quaternion arithmetic in Lipschitz integers to establish new sum representations.

Provides algebraic conditions for restricted sums of four squares.

Abstract

In this paper, we prove some results of restricted sums of four squares using arithmetic of quaternions in the ring of Lipschitz integers. For example, we show that every nonnegative integer $n$ can be written as $x^{2}+y^{2}+z^{2}+t^{2}$ where $x,y,z,t$ are integers and $x+y+2z+2t$ is a square or a cube.