Erdős Problem #649

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 \(q=7\). It is possible that Erdős meant to exclude such obstructions, by amending this to 'odd primes' or 'all sufficiently large primes' or such. The statement below assumes \(p \neq q\): for \(p = q\) no such \(n\) exists, because \(p\) would divide both \(n\) and \(n+1\).