Smallest prime factor of a Mersenne number

Question

The Mersenne numbers $M_n$ are integers of the form $2^n-1$, where $n$ is a positive integer. In the case when $n$ is a prime, are there any results known on the smallest prime factor, $p_n$, of $M_n$, as a function of $n$? I've seen some papers on the largest prime factor, see here for instance

https://www.math.dartmouth.edu/~carlp/PDF/murata4.pdf

but nothing yet on smallest. I'm especially interested in knowing whether $p_n \to \infty$ as $n \to \infty$ for $n$ prime? Thank you.

Answer 1

From Wikpedia: "If p is an odd prime, then every prime q that divides $2^p − 1$ must be 1 plus a multiple of 2p"

So the smallest possible number dividing $2^p − 1$ is $2p+1$ which obviously goes to infinity as p does.

http://en.wikipedia.org/wiki/Mersenne_prime#Theorems_about_Mersenne_numbers

Answer 2

The question seems to be related to the Mersenne conjectures. We restrict ourselfs to $n=p$ prime. If there are infinitely many Mersenne primes $M_p$, then the least prime divisor of $M_p$ would be $M_p$ itself infinitely often. On the other hand, the least prime factor seems to be much smaller than $M_p$ infinitely often, too.

Answer 3

Taking @dioid's comment further, we know that for a prime $p\neq 2$ we have $p|2^{p-1}-1$ so that $p|2^{n(p-1)}-1$

If $q_p$ is the least exponent for which $p|2^{q_p}-1$, then it is easy to show that $p|2^q-1$ only if $q_p|q$

So if $q>q_3, q_5, q_7 \dots q_p$ is a prime, then no prime $\leq p$ can be a factor of $2^q-1$