# The Integer Cuboid Table

## TL;DR

This paper presents an exhaustive computational search for integer cuboids, categorizing them into three types, and compiles a comprehensive table of 167,043 such cuboids with edges up to over 200 billion.

## Contribution

It introduces a systematic search method using the Pythagorean group to identify all integer cuboids within a large range, expanding the known catalog significantly.

## Key findings

- Discovered 167,043 integer cuboids within the specified range.
- Classified cuboids into Euler, edge, and face types based on their properties.
- Provided an extensive table of these cuboids for future research.

## Abstract

Integer cuboids are rectangular Diophantine parallelepipeds It has been discovered that these cuboids come in 3 varieties: Euler or body type, edge type, and face type. In all three cases, one edge or diagonal is irrational, all six others are rational. We discuss an exhaustive computer search procedure which uses the Pythagorean group Py(n) to locate all possible cuboids with a given edge n. Over the range of 44 to 200,000,000,027 for the smallest edge, 167,043 cuboids were discovered. They are listed in the Integer Cuboid Table.