Side lengths of cubes with vertices in $\mathbb Z^n$

TL;DR

This paper characterizes the possible side lengths of d-dimensional cubes whose vertices are integer lattice points in n-dimensional space, using advanced algebraic methods.

Contribution

It introduces a novel application of Witt's cancellation theorem to determine the set of feasible side lengths for such cubes.

Findings

Complete characterization of side lengths for cubes with integer vertices

Application of quadratic form theory to geometric lattice problems

Extension of algebraic methods to higher-dimensional geometric configurations

Abstract

We determine the set of side lengths of $d$-dimensional cubes with vertices in $\mathbb Z^n$ using Witt's cancellation theorem from the algebraic theory of quadratic forms.