An prime number $p$ is called *pathological* if there exists a prime number $q\ne p$ such that for every $n\in\mathbb N$ the number $2^n-1$ is divisible by $p$ if and only if $2^n-1$ is divisible by $q$.  According to the comment of Gerhard Paseman and @YCor to [this problem][1], pathological prime numbers exist and the smallest one is 23, the next is 53, then 89, 157, etc.

>**Problem 1.** What is the Banach density of pathological prime numbers? Is is zero? 

Is a version of the Dirichlet density theorem true for non-pathological prime numbers:

>**Problem 2.** Is it true that for every natural number $a$ and any (square-free) number $b$, which is relatively prime with $a$, the arithmetic progression $a+b\mathbb N=\{a+bn:n\in\mathbb N\}$ contains a non-pathological odd prime number?


  [1]: https://mathoverflow.net/q/347774/61536