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, pathological prime numbers exist and the smallest one is 23, the next is 53, then 89, 157, etc.

Problem 1. How large is the set of non-pathological primes?

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?

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, pathological prime numbers exist and the smallest one is 23, the next is 53, then 89, 157, etc.

Problem 1. How large is the set of non-pathological primes?

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?

Added in Edit. By Bang's Theorem, for any $n\ge 2$, the number $2^n-1$ has a non-pathologic prime divisor. Is this information sufficient for answering Problem 2 in affirmative?

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, 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?

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, pathological prime numbers exist and the smallest one is 23, the next is 53, then 89, 157, etc.

Problem 1. How large is the set of non-pathological primes?

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?