# 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

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $44$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $44$