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An Elementary Obstruction to the Existence of a Perfect Cuboid

Stéphane Yelle

Abstract

We study arithmetic constraints arising from the three faces meeting along the space diagonal of a rectangular cuboid. Using a propagation mechanism along this diagonal, based on the appearance of a minimal odd prime in certain triangular remainders, we derive strong structural restrictions on possible configurations. These constraints induce an infinite descent along the space diagonal, preventing the existence of a compatible integral structure. This approach provides an elementary obstruction to the existence of a perfect cuboid, relying only on divisibility and congruence arguments, and avoiding the use of Gaussian integers or classical quadratic factorizations.

An Elementary Obstruction to the Existence of a Perfect Cuboid

Abstract

We study arithmetic constraints arising from the three faces meeting along the space diagonal of a rectangular cuboid. Using a propagation mechanism along this diagonal, based on the appearance of a minimal odd prime in certain triangular remainders, we derive strong structural restrictions on possible configurations. These constraints induce an infinite descent along the space diagonal, preventing the existence of a compatible integral structure. This approach provides an elementary obstruction to the existence of a perfect cuboid, relying only on divisibility and congruence arguments, and avoiding the use of Gaussian integers or classical quadratic factorizations.
Paper Structure (10 sections, 9 theorems, 48 equations, 1 figure)

This paper contains 10 sections, 9 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Propagation Along the Common Space Diagonal
  3. The Triangular Remainder Language
  4. Parametrization by triangular remainders
  5. Minimal choice of an odd prime
  6. Internal propagation within a remainder triangle
  7. Additive transport via the space diagonal
  8. Fundamental trigger
  9. Additive lock and propagation on a face
  10. Final descent

Key Result

Lemma 1

Let with $a,b,c>0$ forming a triangle. Define the triangular remainder by Then $r$ is even.

Figures (1)

  • Figure 1: Cuboïde $ABCDEFGH$ : $AB=a$, $AD=b$, $AE=c$ ; diagonales de faces $AC=d_{ab}$, $AF=d_{ac}$, $AH=d_{bc}$ ; diagonale de l’espace $AG=d_{sp}$.

Theorems & Definitions (19)

  • Lemma 1: Parity of the triangular remainder
  • proof
  • Lemma 2: Fundamental identity of the triangular remainder
  • proof
  • Lemma 3: Parity of $x$ and $y$
  • proof
  • proof
  • Lemma 4: Minimal choice on the remainders
  • proof
  • Lemma 5: Min--2
  • ...and 9 more