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.
Best Answer
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