Does there exist a positive integer $m$ and two increasing positive integer sequences $\left\{ a_n \right\}$ and $\left\{ b_n \right\}$ so that the set $$M=\left\{ p| p\; \textrm{is a prime}, \exists i,j\in \mathbb{N}^+ \; \textrm{so that} \; p | a_ib_j+m \right\}$$ is a finite set?
The question looks interesting, I don't know if it's a famous one, but I guess it doesn't exist, but ..