Added Power
How many integers $n$ are there for which $n$ and $n^3+3$ are both prime?
Age
14 to 16
Challenge level
Problem
For how many primes $p$ is the number $p^3+3$ also prime?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Answer: one
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.