SOLVED
Let $P(n)$ denote the largest prime factor of $n$. There are infinitely many $n$ such that $P(n)>P(n+1)>P(n+2)$.
Conjectured by Erdős and Pomerance
[ErPo78], who proved the analogous result for $P(n)<P(n+1)<P(n+2)$. Solved by Balog
[Ba01], who proved that this is true for $\gg \sqrt{x}$ many $n\leq x$ (for all large $x$).
