Problem #372: Let $P(n)$ denote the largest prime factor of $n$

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Progress

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 $x$). Balog suggests the natural conjecture that the density of such $n$ is $1/6$. A generalised form of this conjecture was presented by De Koninck and Doyon [DeDo11].