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Claim. Theorem 1.1 of Hung M. Bui, Kyle Pratt and Alexandru Zaharescu, Power savings for counting solutions to polynomial-factorial equations, Adv. Math. 422 (2023), Paper No. 109021, 32 pp., states that for a fixed polynomial $P\in\mathbb Z[X]$ of degree $r\ge2$ and a fixed nonzero integer $s$ there is a constant $C$ with

$$\#\{N\le n<2N:\ s\cdot n!=P(x)\text{ for some }x\in\mathbb Z\}\le C\,N^{33/34}.$$

Remark 1.4 and Proposition 3.2 say that the method gives the exponent $12\sqrt2-16+\varepsilon=0.97056\ldots$ (Proposition 3.2 states it as the hypothesis $\theta\le17-12\sqrt2-\varepsilon$), which $33/34=0.97058\ldots$ approximates. The proof splits the solutions into $(2r+1)$-tuples, finds by pigeonhole three solutions with small gaps in one residue class modulo $r$, turns each such triple into a simultaneous rational approximation to values of algebraic functions, in the manner of Berend and Osgood, and bounds the number of such approximations by Diophantine and Padé approximation. The source card is Bui, Pratt and Zaharescu 2023; the arXiv posting is the preprint link.

Consequence for the problem. If $f(n)=m$ in Problem 393, then $n!=\prod_{s\in S}(a+s)$ for some $a\ge1$ and some $S\subseteq\{0,\ldots,m\}$ containing $0$ and $m$, so $n!=P_S(a)$ for one of the finitely many polynomials $P_S(X)=\prod_{s\in S}(X+s)$, each of degree $|S|\ge2$. Summing the theorem over these polynomials and over dyadic ranges gives, with $F_m(N)$ the number of $n\le N$ with $f(n)=m$, $F_m(N)\ll_m N^{33/34}$, as the site's remarks state; this sharpens the $o(N)$ of Berend and Osgood 1992.

Covers. The counting bound only: $F_m(N)\ll_m N^{33/34}$ for each fixed $m$. It does not settle whether $f(n)=1$ infinitely often, which is open unconditionally, nor the growth of $f(n)$ along every $n$.

Acceptance. Refereed: Advances in Mathematics, volume 422 (June 2023), article 109021; the Crossref record of the DOI gives these data. The site labels the problem OPEN, so its remark crediting the result is commentary on an open problem and not acceptance, and no reviewed evidence is listed. The proof is not checked here.

Depends on. No page of this wiki.