Challenge Level

If $p$ is odd, then $p^3$ is odd, so $p^3+3$ is even, and therefore not prime (as it is greater than $2$).

If $p$ is $2$, then $p^3+3=11$, which is prime.

So for only one prime $p$, $p^3+3$ is also prime.

This problem is taken from the UKMT Mathematical Challenges.

You can find more short problems, arranged by curriculum topic, in our short problems collection.