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Source. Proposition 1, p. 270, 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, as identified on the source card.

Read depth. Claims checked: the statement, the setting of equation (1) on p. 269 and the form of the abc conjecture on p. 270 were read clause by clause on the print; the proof (pp. 270--273) was read for structure only. Nothing here is independently reviewed.

Statement

Setting (p. 269). $P\in\mathbb Z[X]$ is any polynomial with integer coefficients of degree $d\ge2$, and equation (1) is $P(x)=n!$ with $x$ an integer.

Proposition 1 (p. 270). "The ABC-conjecture implies that equation (1) has only finitely many solutions $(x,\,n)$."

The sentence introducing the proposition on the same page, and the abstract on p. 269, state the conclusion as finitely many integer solutions $(x,n)$ with $n>0$, for an arbitrary $P$ of degree $d\ge2$.

Hypothesis (p. 270). The abc conjecture in the form the paper uses: for every $\varepsilon>0$ there is a constant $C(\varepsilon)$, depending only on $\varepsilon$, such that any three coprime nonzero integers $A,B,C$ with $A+B=C$ satisfy $\max(|A|,|B|,|C|)<C(\varepsilon)\,N(ABC)^{1+\varepsilon}$, where $N(k)=\prod_{p\mid k}p$ is the radical of a nonzero integer $k$. The proof applies it with the single value $\varepsilon=1/(2d)$.

Proof pointer

Pages 270--273. Multiplying (1) by $d^da_0^{d-1}$, where $a_0$ is the leading coefficient of $P$, and shifting the variable gives a monic equation $Q(z)=c\,n!$ with $Q(X)=X^d+R(X)$ and no $X^{d-1}$ term, where $c=d^da_0^{d-1}$; for large $|z|$, $d\log|z|$ and $\log n!$ differ by a bounded amount. If $R=0$, a prime in $(n/2,n)$ larger than $c$ divides $c\,n!$ exactly once when $n>2c$, so $c\,n!$ is not a $d$th power. Otherwise, after removing the power of $z$ dividing $R$ and a common factor, the abc conjecture applied to the resulting three-term equation, with the radical of $c\,n!$ bounded by $\prod_{p\le n}p<4^n$, gives $\log|z|\ll n$; with Stirling's formula this bounds $n$, and then $|z|$. Not checked here.

Dependencies

The abc conjecture, unproved, as above; the elementary bound $\prod_{p\le n}p<4^n$, Stirling's formula, and a prime in $(n/2,n)$. The paper generalizes Overholt's theorem that a weak form of abc (a constant $e>0$ with $|x^3-y^2|<N(x^3-y^2)^e$ for all integers $x,y$ with $x^3\ne y^2$) gives finitely many solutions of $x^2-1=n!$ (p. 270, citing Overholt, Bull. London Math. Soc., 1993).

Bears on

• Problem 393: if $f(n)=m$, then $n!=P_S(a)$ for some $a\ge1$ and one of the finitely many polynomials $P_S(X)=\prod_{s\in S}(X+s)$, $S\subseteq\{0,\ldots,m\}$ containing $0$ and $m$, each of degree $|S|\ge2$; under the abc conjecture the proposition gives finitely many $n$ with $f(n)=m$ for each $m$, so $f(n)\to\infty$. This reduction is the problem page's, not the paper's, which names no such $f$. The result is conditional on abc and says nothing about the rate of growth of $f(n)$.
• Problem 398: the case $P(X)=X^2-1$ gives, under the abc conjecture, only finitely many $n$ with $n!=x^2-1$; it does not show that the solutions are only $n=4,5,7$, and it is conditional on abc.