The Euler Brick/Perfect Cuboid Problem

 I've mentioned before that there are a lot of simple to state open math problems that are not so famous. Let's talk about another of them: The perfect cuboid problem, sometimes called the Euler brick problem.

 

The Euler brick problem/perfect cuboid problem is to find an example of a rectangular box such that every side length, each face diagonal, and the internal diagonals are integers. We call such an object a perfect cuboid.

 

We have examples of where all sides and face diagonals are integers. An example is a box with width 44, length 117, and height 240. But there no known example where every face and the internal diagonals are integers.

 

In 2009, Jorge Sawyer and Clifford Reiter discovered examples of parallelepipeds (that is a box with parallelograms for sides) where are all sides, face diagonals and internal diagonals, are integers.

 

There are a lot of restrictions on what a perfect cuboid must look like. For example, we know that the smallest side must be at least 10^10. This number has been claimed to be pushed up the last few years but those higher searches have not yet been peer reviewed as far as I'm aware. One natural thing one can try to prove that the sides must be divisible by various primes. For example, one can without too much work show that at least one of the edge's must be divisible by 5. More generally, if one lets p be any prime of the set 2, 3, 5, 7, 11, 13, 17, 19, 29, 37 then at least one edge, face diagonal or internal diagonal must be divisible by p. You may note that the list above skips 23 and 31; the obvious sort of arguments don't seem to work for those two. Extending this to larger primes likewise seems difficult.