Although I am also interested in the number of distinct prime factors (not countingmultiplicity), today I use $\omega(m)$ to denote the number of (positive) primefactors (with multiplicity) of the integer $m$. Thus $\omega(75)=3$ in this post.(I may switch to $\omega(75)$ being 2 in a different post.)
What is known about $\omega(p^n - 1)$ for fixed integer $p \gt 1$ and growing $n$?When $n$ is composite, algebraic factorization guarantees something like $\Omega(\omega(n))$factors. I am especially interested in cases where $n\lt \omega(p^n - 1)$. I do not havea proof, but I think that for fixed $p$ one can show there are only finitely many suchcases.
If something is known for $p$ prime, that would interest me greatly. I still thinkthe general case is of note, and would appreciate a reference.
