Cuboid with Integer Edges and Face Diagonals

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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.


Sources