# Prime Equal to Sum of Digits of Cube

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## Theorem

The only prime number which is equal to the sum of the digits of its cube is $17$:

## Proof

We have that:

\(\displaystyle 17^3\) | \(=\) | \(\displaystyle 4913\) | |||||||||||

\(\displaystyle 17\) | \(=\) | \(\displaystyle 4 + 9 + 1 + 3\) |

From Positive Integers Equal to Sum of Digits of Cube, the complete set of positive integers with this property are:

- $0, 1, 8, 17, 18, 26, 27$

Of these, only $17$ is prime.

$\blacksquare$

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $17$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $17$