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.
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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): $44$