Prime Dudeney Number

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

Proof
We have that:

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.