Some Connections Between The Arithmetic and The Geometry of Lipschitz Integers

TL;DR

This paper explores the relationship between the arithmetic and geometric properties of Lipschitz and Hurwitz integers, revealing new algebraic identities and posing geometric problems related to quaternion factorization.

Contribution

It establishes novel connections between vector products and multiples of Lipschitz integers, and offers new arithmetical proofs for classical results.

Findings

Vector product of Lipschitz integers yields multiples of the original integer.

Certain vector products are both left and right multiples, indicating symmetry.

Raises a geometric problem related to quaternion factorization and integer location.

Abstract

Some relationships between the arithmetic and the geometry of Lipschitz and Hurwitz integers are presented. In particular, it is shown that the (ternary) vector product of a Lipschitz integer $\alpha$ with two other Lipschitz integers, both orthogonal to $\alpha$, is a left and also a right multiple of $\alpha$, and that the vector product of two left multiples of $\alpha$ with any other Lipschitz integer is still a left multiple of $\alpha$. We also provide new arithmetical proofs for some old results of Gordon Pall, and raise a geometric problem on the location of some integral quaternions that is related to the factorization of some integers.