# Cuboid with Integer Edges and Face Diagonals

## Theorem

The smallest cuboid whose edges and the diagonals of whose faces are all integers has edge lengths $44$, $117$ and $240$.

Its space diagonal, however, is not an integer.

## Proof

The edges are given as having lengths $44$, $117$ and $240$.

The faces are therefore:

$44 \times 117$
$44 \times 240$
$117 \times 240$

The diagonals of these faces are given by Pythagoras's Theorem as follows:

 $\ds 44^2 + 117^2$ $=$ $\ds 15 \, 625$ $\ds$ $=$ $\ds 125^2$

 $\ds 44^2 + 240^2$ $=$ $\ds 59 \, 536$ $\ds$ $=$ $\ds 244^2$

 $\ds 117^2 + 240^2$ $=$ $\ds 71 \, 289$ $\ds$ $=$ $\ds 267^2$

However, its space diagonal is calculated as:

 $\ds 44^2 + 117^2 + 240^2$ $=$ $\ds 73 \, 225$ $\ds$ $=$ $\ds \paren {270 \cdotp 6012 \ldots}^2$

which, as can be seen, is not an integer.

## Historical Note

This result was discovered by Leonhard Paul Euler.

However, he was unable to find such a cuboid whose space diagonal is also an integer.