The density of the set of non-pathological primes

Question

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?

Answer 1

One can phrase this is terms of divisors of (values when evaluated at $x=2$ of) cyclotomic polynomials $P_n(x)$. A pathological prime $p$ is a primitive prime divisor of $P_n(2)$ (so $p$ does not divide any $P_m(2)$ for $m$ smaller than $n$) such that there is $q$ different from $p$ which is also a primitive prime divisor of $P_n(2)$. Given the sparsity of Mersenne primes, one can find many pathological primes as divisors of Mersenne non-primes but with the exponent of this Mersenne number also being prime. So barring exceptional behaviour of the sequence of Mersenne primes, we have a potential source of infinitely many pathological prime numbers: just look at almost any Mersenne number with prime exponent ($P_n(2)$ is a Mersenne number when $n$ is prime).

When $n$ is not prime, $P_n(2)$ may yield some more pathological primes, unless it is a nontrivial power of a prime. (For $n$ bigger than 6, there will be at least one primitive prime divisor; Zsigmondy and Bang showed this, and Granville had something to add more generally about the parity of the exponent of such primes.) However, there are some $n$ for which $P_n(2)$ is prime, and these provide nonpathological primes. Further, these primes are 1 mod n, and so knowing which $n$ these are give a start on approaching Problems 1 and 2. Looking at Cunningham tables suggests that these primes are frequent in occurrence and possibly also infinite in number, regardless of the infinitude of Mersenne primes (which are themselves nonpathological).

Update 2019.12.12:

After rereading notes of G. Jameson on cyclotomic polynomials (see comments below for an indirect reference), we can be more clear on the presence of nonpathologic primes. Although one can look at P_n(a) for values of a other than 2, I stick with Taras's choice of 2. Further, for n less than 7, one can study those numbers individually, so I will speak of n bigger than 6.

Prime divisors of P_n(2) are bigger than n, except in one case: if n has a special form mq^k where q is the largest prime divisor of n and q is a primitive divisor of P_m(2), then q can also divide P_n(2). So there are two possibilities for p not to be a pathological prime: if n is not special and P_n(2) has one prime divisor (and I would expect all but finitely many to not be proper prime powers, but this is unproven), or if n is special and P_n(2) is qp^k (less is known about this, but k bigger than 1 would surprise me).

I don't know how to apply this to Will Sawin's comment. This form suggests to me that maybe the upper bound on the number of non pathological primes below X could actually be log of his suggested upper bound. If indeed there are infinitely many, then each will be 1 mod n for some n, and when n is sufficiently large may fall into enough of the arithmetic progressions of part 2.

End Update 2019.12.12.

Gerhard "Tables Nice Place To Frequent" Paseman, 2019.12.09.