Let $P$ denote the set of primes in $\mathbb{N}$. For $k\in \mathbb{N}, k\geq 2$ set $$M\_k = \big\{p\in P: \{kp-1, kp+1\}\cap P \neq \emptyset\big\}.$$
Is there $k\in \mathbb{N}, k\geq 2$ such that $M\_k$ is infinite?



Primes that are "almost" multiples of each other
You're asking whether, for some $k \ge 2$, there are infinitely many primes $p$ such that either $kp-1$ or $kp+1$ is prime. That would mean there are infinitely many primes such that $kp-1$ is prime or infinitely many primes such that $kp+1$ is prime. Of course, $k$ had better be even. If $k$ is even, Dickson's conjecture would say that in both cases the answer is yes.  

But Dickson's is still a conjecture, and there are no $k$ for which the answer has been proven to be yes. 


