OPEN

Is it true that, for any two primes $p,q$, there exists some integer $n$ such that the largest prime factor of $n$ is $p$ and the largest prime factor of $n+1$ is $q$?

Erdős writes 'it is probably hopelessly difficult to decide about the truth of this conjecture'. The number of solutions is finite for any fixed $p,q$ since the largest prime factor of $n(n+1)$ tends to $\infty$ (Mahler [Ma35] showed that this is $\gg \log\log n$, see [368]).

More generally, one can ask about whether for any primes $p_1,\ldots,p_k$ there exists some $n$ such that the largest prime factor of $n+i$ is $p_i$. Erdős writes this is 'clearly impossible' if the $p_i$ are the first $k$ primes and $k$ is sufficiently large, but does not know what happens if all of the primes are sufficiently large compared to $k$.