Let both $p$ and $k$ be prime numbers such that $(p^2 + k)$ is also a prime number less than 30. What is the…
Let both $p$ and $k$ be prime numbers such that $(p^2 + k)$ is also a prime number less than 30. What is the number of possible values of $k$?
4
5
6
7
Solution
For $p^2+k$ to be a prime less than 30, $p$ must generally be 2 (if $p$ is odd, $p^2$ is odd and $k$ prime is mostly odd, making the sum even). With $p=2$, $p^2=4$, so $4+k$ must be prime and less than 30: $k=3\Rightarrow7$, $k=7\Rightarrow11$, $k=13\Rightarrow17$, $k=19\Rightarrow23$ work; with $p=3, k=2\Rightarrow11$ also works. Counting the distinct valid $k$ values gives 5.