There is no Perfect Cuboid

TL;DR

This paper proves that an Euler brick cannot have an odd internal diagonal, providing a simple proof that no perfect cuboid exists, thus resolving a long-standing open problem in number theory.

Contribution

The paper offers a straightforward proof demonstrating the impossibility of an Euler brick having an odd internal diagonal, ruling out the existence of a perfect cuboid.

Findings

Proves the internal diagonal of an Euler brick cannot be odd.

Shows no perfect cuboid exists based on the internal diagonal property.

Simplifies the understanding of the perfect cuboid problem.

Abstract

A perfect cuboid is formed when an Euler brick whose edges and face diagonals are all integers also has an integer internal diagonal. It is known that if a perfect cuboid exists the internal diagonal is odd. No perfect cuboid has been found. This simple proof shows that the internal diaogonal of an Euler brick cannot be an odd integer.