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Claim. Proposition 1 of Florian Luca, The Diophantine equation $P(x)=n!$ and a result of M. Overholt, Glas. Mat. Ser. III 37(57) (2002), no. 2, 269--273, states that the abc conjecture implies that, for every polynomial $P\in\mathbb Z[X]$ of degree $d\ge2$, the equation $P(x)=n!$ with $x$ an integer has only finitely many solutions $(x,n)$; the sentence introducing it and the abstract say integer solutions $(x,n)$ with $n>0$. The proof multiplies through to a monic equation $z^d+c_2z^{d-2}+\cdots+c_d=c\,n!$, and compares the size and the radical of the terms of that equation, which the abc conjecture bounds, with the growth of $n!$. The paper generalizes Overholt's result that a weak form of abc gives finiteness for the Brocard--Ramanujan equation $x^2-1=n!$. The source card is Luca 2002.

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 $|S|\ge2$. Under abc each of these equations has finitely many solutions, so for each $m$ only finitely many $n$ have $f(n)=m$; that is, $f(n)\to\infty$, and in particular $f(n)=1$ only finitely often, which answers with no the question of Erdős and Graham whether a factorial is the product of two consecutive integers infinitely often. The site's remarks record this consequence.

Hypothesis. The claim is conditional on the abc conjecture: for every $\varepsilon>0$ there is a constant $C(\varepsilon)$ such that coprime nonzero integers $A+B=C$ satisfy $\max(|A|,|B|,|C|)<C(\varepsilon)\,\mathrm{rad}(ABC)^{1+\varepsilon}$. The conjecture is unproved, so this page derives nothing for the problem's standing; whether $f(n)=1$ infinitely often is open unconditionally.

Acceptance. Refereed: Glasnik Matematički, Series III, volume 37(57), number 2 (2002), pp. 269--273, as the journal's article page records. The site labels the problem OPEN, so its remark that Luca's result implies $f(n)\to\infty$ under abc 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.