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

Christian Bernert, Jens Reinhold

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.

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

Abstract

We determine the set of side lengths of -dimensional cubes with vertices in using Witt's cancellation theorem from the algebraic theory of quadratic forms.
Paper Structure (3 sections, 4 theorems, 9 equations, 1 figure)

This paper contains 3 sections, 4 theorems, 9 equations, 1 figure.

Table of Contents

  1. Background
  2. A geometric generalization
  3. Proofs

Key Result

Theorem 1

For positive integers $d \leq n$, the set $J(d,n)$ is given by the entries of the following table.

Figures (1)

  • Figure 1:

Theorems & Definitions (8)

  • Theorem 1
  • Remark
  • Lemma 2: Witt's cancellation theorem
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Remark