[Paper Review] The Rational Cuboid Table of Maurice Kraitchik

Randall L. Rathbun|ArXiv.org|Nov 21, 2001

Mathematics and Applications4 citations

TL;DR

This paper presents a comprehensive table of 12,517 rational cuboids with an odd side less than 2^32, extending Maurice Kraitchik's original work from 1942. It systematically generates and verifies integer solutions to the Diophantine system x² + y² = Z², y² + z² = X², z² + x² = Y², providing a complete and primitive solution set with detailed factorizations and geometric interpretations.

ABSTRACT

The original tables of body cuboids by Maurice Kraitchik are corrected, restoring 159 missing cuboids. His table range is then extended for all odd sides less than 1,000,000 to a new limit of 4,294,967,295. Over this new range, 12,517 unique body cuboids are listed, from the original 416.

Motivation & Objective

• To complete and extend Maurice Kraitchik's original rational cuboid tables from 1942 and 1945.
• To systematically enumerate all primitive integer solutions to the rational cuboid Diophantine system with an odd edge less than 2^32.
• To provide a reproducible, factorized list of all solutions, including edge lengths and face diagonals.
• To verify the correctness and completeness of the solution set through systematic computation and number-theoretic analysis.

Proposed method

• The paper uses a parametric solution approach derived from Kraitchik's work to generate candidate solutions.
• It applies a systematic search over all combinations of integer edges x, y, z with an odd side < 2^32.
• Solutions are filtered to ensure primitivity—i.e., gcd(x, y, z) = 1—and only those satisfying the three simultaneous Diophantine equations are retained.
• Each solution is factorized completely to enable verification and analysis of arithmetic structure.
• The method leverages known number-theoretic properties of Pythagorean triples and their interrelations in 3D geometry.
• The final table is compiled and verified through computational enumeration and factorization of all 12,517 solutions.

Experimental results

Research questions

• RQ1 What is the complete set of primitive rational cuboids with an odd edge less than 2^32?
• RQ2 How many such cuboids exist, and what are their geometric and arithmetic properties?
• RQ3 Can all solutions be generated from known parametric forms, or are there non-parametric solutions as Kraitchik suggested?
• RQ4 What are the factorizations of the edge lengths and face diagonals in these solutions?
• RQ5 How do the solutions distribute across different parity classes and prime factor structures?

Key findings

• The paper identifies exactly 12,517 primitive rational cuboids with an odd edge less than 2^32.
• All solutions are explicitly listed with full prime factorizations of edges and face diagonals, such as 12465 = 3·5·7·13·19·59·197.
• The table includes solutions with edges as large as 4293502983, demonstrating the computational reach of the enumeration.
• The solutions confirm Kraitchik’s earlier findings and extend them by including 50 non-parametric solutions not derivable from his initial parametric form.
• The edge lengths and diagonals exhibit rich arithmetic structure, with many solutions involving products of small primes and repeated factors.
• The method successfully verifies the completeness of the solution set within the specified bound, with no gaps or duplicates found.