Finding Sums of Four Squares via Complex Continued Fractions

TL;DR

This paper presents a novel method using complex continued fractions, specifically Hurwitz algorithm, to find representations of odd integers as sums of four squares, avoiding quaternion algebra complexities.

Contribution

It introduces a new approach leveraging complex continued fractions to find four-square representations, simplifying previous quaternion algebra methods.

Findings

Avoids quaternion algebra complexities

Uses Hurwitz algorithm for complex expansion

Provides an alternative to existing methods

Abstract

The problem of representing a given positive integer as a sum of four squares of integers has been widely concerned for a long time, and for a given positive odd $n$ one can find a representation by doing arithmetic in a maximal order of quaternion algebra once a pair of (positive) integers $x,y$ with $x^2+y^2\equiv-1\mod n$ is given. In this paper, we introduce a new method to find a representation of odd integer $w$ given $x,y$ satisfying the above requirement. This method can avoid the complicated non-commutative structure in quaternion algebra, which is similar to the one we use to obtain a representation of a prime $p\equiv1\mod4$ as sum of two squares by doing continued fraction expansions, except that here we will expand complex number using Hurwitz algorithm.