Sums of squares and orthogonal integral vectors

TL;DR

This paper characterizes twin pairs of orthogonal integral vectors in three dimensions using cubic lattices, introduces the concept of twin-complete integers, and relates their properties to a major number theory conjecture, employing quaternion algebra.

Contribution

It provides a complete characterization of twin-complete integers and connects their properties to a well-known number theory conjecture, using quaternion decomposition techniques.

Findings

Characterization of twin pairs via cubic lattices

Definition and analysis of twin-complete integers

Connection to a famous number theory conjecture

Abstract

Two vectors in $\BZ^3$ are called \emph{twins} if they are orthogonal and have the same length. The paper describes twin pairs using cubic lattices, and counts the number of twin pairs with a given length. Integers $M$ with the property that each integral vector with length $\sqrt{M}$ has a twin are called twin-complete. They are completely characterized modulo a famous conjecture in number theory. The main tool is the decomposition theory of Hurwitz integral quaternions. Throughout the paper we made a concerted effort to keep the exposition as elementary as possible.