OPEN

Let $P(n)$ denote the largest prime factor of $n$. Show that the set of $n$ with $P(n+1)>P(n)$ has density $1/2$.

Conjectured by Erdős and Pomerance

[ErPo78], who proved that this set and its complement both have positive upper density.

In [Er79e] Erdős also asks whether, for every $\alpha>0$, the density of the set of $n$ where
\[P(n+1)>P(n)n^\alpha\]
exists.