Euler Brick - Perfect Cuboid

A perfect cuboid (also called a perfect box) is an Euler brick whose space diagonal also has integer length.

In other words, the following equation is added to the system of Diophantine equations defining an Euler brick:

As of November 2012, no example of a perfect cuboid had been found and no one has proven that none exist. Exhaustive computer searches show that, if a perfect cuboid exists, one of its edges must be greater than 3·1012. Furthermore, its smallest edge must be longer than 1010.

Some facts are known about properties that must be satisfied by a primitive perfect cuboid, if one exists:

• 2 of the edges must have even length and 1 edge must have odd length
• 2 edges must have length divisible by 4 and at least 1 of those edges must have length divisible by 16
• 2 edges must have length divisible by 3 and at least 1 of those edges must have length divisible by 9
• 1 edge must have length divisible by 5
• 1 edge must have length divisible by 7
• 1 edge must have length divisible by 11
• 1 edge must have length divisible by 19.

Solutions have been found where the space diagonal and two of the three face diagonals are integers, such as:

Solutions are also known where all four diagonals but only two of the three edges are integers, such as:

and