# Does there exist a positive integer $m$ and two increasing positive integer sequences is a finite set?

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 ..

• @lulu sorry, I misread, I'll delete it – Mark Bennet May 25 '18 at 15:10
• Assuming the abc conjecture, they don't exist. Short of that, I can't be sure. – Ivan Neretin May 30 '18 at 19:06
• What's $m{}{}{}$? – GFauxPas May 31 '18 at 0:44
• Interesting question indeed! If I may ask, how did this question arise? My first thought was to show that for just one strictly increasing sequence but unfortunately that case is trivial and has no room for generalization... – Μάρκος Καραμέρης Jun 5 '18 at 22:09