Problem #393: Let $f(n)$ denote the minimal $m\geq 1$ such that\[n! =...

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Progress

Erdős and Graham write that they do not even know whether $f(n)=1$ infinitely often (i.e. whether a factorial is the product of two consecutive integers infinitely often).

Let $F_m(N)$ count the number of $n\leq N$ such that $f(n)=m$. Berend and Osgood [BeOs92] proved that, for each fixed $m$, $F_m(N)=o(N)$. Bui, Pratt, and Zaharescu [BPZ23] have shown that\[F_m(N)\ll_m N^{33/34}.\]A result of Luca [Lu02] implies that $f(n)\to \infty$ as $n\to \infty$, conditional on the ABC conjecture.