Let $n\geq 1$ an integer, in this post we denote the greatest prime dividing $n$ as $\operatorname{gpf}(n)$.
See it you want the article from MathWorld Greatest Prime Factor.
While I was writing equations and congruence relations involving different arithmetic functions and $\operatorname{gpf}(n)$ I wondered about if there exist infinitely many primes of the form $$2\cdot n^{\operatorname{gpf}(n)}+1.\tag{1}$$
I was doing experiments with different factors instead our first factor $2$. Now I have no intuition if should have infinitely many primes of the form $(1)$.
Question. What work can be done about the existence of infinitely many primes of the form $$2\,m^{\operatorname{gpf}(m)}+1$$ when $m\geq 1$ runs over positive integers? Or well, can you provide a heuristic or reasoning whereby we should think that there exist only a finite number of them? Many thanks.
Trivially for every prime $n$, $gpf(n)=n$. So your conjecture or question is the same as the Sophie Germain prime conjecture, which says infinitely many primes $n$ such that $2n+1$ is also prime. This has yet to be proven.