Let $P(m)$ denote the greatest prime factor of $m$. Is it true that, for any two primes $p,q$, there exists some integer $n$ such that $P(n)=p$ and $P(n+1)=q$?

In fact, the answer to this question as written is easily seen to be no, since there are no solutions to $2^k\equiv -1\pmod{7}$, and hence this fails with $p=2$ and $q=7$.