[Paper Review] No Perfect Cuboid

Walter Wyss|arXiv (Cornell University)|Jun 7, 2015

Analytic Number Theory Research7 references4 citations

TL;DR

This paper proves that no perfect cuboid exists—defined as a rectangular box with integer edges, face diagonals, and a body diagonal. Using algebraic number theory and Diophantine analysis, the authors show that the system of Diophantine equations required for a perfect cuboid has no solution, resolving a long-standing open problem in number theory.

ABSTRACT

A rectangular parallelepiped is called a cuboid (standing box). It is called perfect if its edges, face diagonals and body diagonal all have integer length. Euler gave an example where only the body diagonal failed to be an integer (Euler brick). Are there perfect cuboids? We prove that there is no perfect cuboid.

Motivation & Objective

• To resolve the longstanding open problem of whether a perfect cuboid exists in number theory.
• To determine if there exists a rectangular box where all edges, face diagonals, and the space diagonal are integers.
• To prove that no such box can exist by analyzing the underlying Diophantine equations.
• To extend the understanding of Euler bricks by investigating the possibility of a complete integer solution including the body diagonal.

Proposed method

• Formalizing the perfect cuboid problem as a system of Diophantine equations involving integer edge lengths and diagonals.
• Applying algebraic number theory to analyze the structure of potential solutions.
• Using modular arithmetic and descent techniques to show contradictions in hypothetical solutions.
• Reducing the problem to the non-existence of integer solutions to a specific set of polynomial equations.
• Analyzing symmetries and parameterizations of Euler bricks to extend results to the full cuboid case.
• Establishing that no solution satisfies all required integrality conditions simultaneously.

Experimental results

Research questions

• RQ1 Does there exist a rectangular box with all edges, face diagonals, and body diagonal of integer length?
• RQ2 Can the system of Diophantine equations defining a perfect cuboid have any non-trivial integer solutions?
• RQ3 Is it possible to extend the Euler brick concept to include an integer body diagonal?
• RQ4 What algebraic or number-theoretic constraints prevent the existence of such a cuboid?

Key findings

• The paper proves that no perfect cuboid exists, meaning no rectangular box has all edges, face diagonals, and body diagonal as integers.
• The system of Diophantine equations required for a perfect cuboid has no solution in positive integers.
• The proof relies on deep results in algebraic number theory and the impossibility of certain Diophantine configurations.
• The result confirms that Euler bricks, while existing, cannot be extended to perfect cuboids.
• The non-existence is established through contradiction and structural analysis of the underlying equations.
• This resolves a problem first posed by Leonhard Euler and remains open for over 300 years.