Does there exist a rectangular box with positive integer edge lengths a,b,c
such that all three face diagonals and the space diagonal are integers?
a^2+b^2, a^2+c^2, b^2+c^2, a^2+b^2+c^2
must all be perfect squares.
The problem is a concrete meeting point of Pythagorean triples, rational points on algebraic varieties, and computational Diophantine search.
Boxes with integer edges and two integer face diagonals exist. No perfect cuboid is known, and no proof of nonexistence is known.
The four square conditions are strongly coupled. Parameterising one or two Pythagorean relations generally makes the remaining conditions harder.
Parameterise primitive Pythagorean triples, construct an Euler brick, then write the remaining space-diagonal requirement as a rational-point problem.
After scaling by one edge, the problem becomes one about rational points on an intersection of quadrics. This explains why elementary parametrisations make progress but tend to create new square conditions rather than remove them.
For every claimed cuboid, check all four squares independently and verify that all lengths are positive integers. A rational cuboid is not automatically an integer one; clearing denominators changes the scale but must preserve every diagonal condition.
Study Pythagorean triples, Euler bricks, rational points on conics, and the relation between simultaneous square conditions and elliptic curves.