What can be said about the following question?
For which prime numbers $p$ there exists another prime number $q(p)$ such that $q(p)$ does not divide $\lambda p +1$ for all integers $\lambda$ ?
For example, $2$ is a prime number not satisfying the above condition. We also know there are infinite prime numbers in the set $\{\lambda p +1 | \lambda \in Z\}$ by Dirichlet's theorem.
Does anybody know a simple answer, or any reference talking about this kind of question?