# Prime Equal to Sum of Digits of Cube

## 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.

$\blacksquare$