Sums of squares in Quaternion rings

TL;DR

This paper explores how integers can be expressed as sums of squares within Quaternion rings, extending classical number theory results to a non-commutative algebraic setting.

Contribution

It determines the minimal number of squares needed for infinitely many Quaternion rings and provides global bounds, advancing understanding of sums of squares in algebraic structures.

Findings

Identifies the minimum number of squares needed in Quaternion rings

Provides global upper and lower bounds for sums of squares

Extends classical number theory results to Quaternion rings

Abstract

Lagrange's Four Squares Theorem states that any positive integer can be expressed as the sum of four integer squares. We investigate the analogous question over Quaternion rings, focusing on squares of elements of Quaternion rings with integer coefficients. We determine the minimum necessary number of squares for infinitely many Quaternion rings, and give global upper and lower bounds.