Prove or disprove: there exists $a ∈ \Bbb N$ such that for all $n ∈ \Bbb N$, $an + 1$ is prime.

Question

I'm trying to do this homework problem that states the following:

Prove or disprove: there exists $a \in \Bbb N$ such that for all $n \in \Bbb N$, $an + 1$ is prime.

I tried splitting it into cases based on the parity of $a$ and $n$ but so far all I've really been able to prove is the pretty obvious part that $an+1$ can't be even.

Any hints on how to solve this?

Answer 1

Hint: For any $a$, we have $(2a-1)^2 = a(4a-4) + 1$