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.

This article is complete as far as it goes, but it could do with expansion.
In particular: Add the definition of Definition:Euler Brick, and rewrite and rename as appropriate.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Expand}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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.

This theorem requires a proof.
In particular: Still to be proved that this is the smallest.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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): $44$