An Euler brick is a right rectangular prism with integer edge lengths, such that the diagonal of each face is also of integer length. A perfect Euler brick would be one whose space diagonal (the line between opposite vertices of the prism) is also of integer length, but no such brick has been discovered.

Equivalently, are there positive integers a, b, and c such that the Pythagorean sum of any pair of them is an integer, and the Pythagorean sum of all three is also an integer?