Let $P(x) \in \mathbb Z[x]$ of degree $\ge 2$ and $S = \{(x,n) \in \mathbb Z^{2}:P(x)=n!\}$. It is known:
$(a)$ $|S| < \infty$ under the $abc$ conjecture.
$(b)$ Density of $S$ in $\mathbb Z$ is $0$ unconditionally.
Fix $k \in \mathbb Z$. Let:
$(1)$ $P[k] = \{P(x) = \displaystyle \sum_{i=0}^{t}p_{i}x^{i} \in \mathbb Z[x]$: $p_{i}$ ~ $O(k^{c})$ $\forall i$, $t$ ~ $O(\log^{d} {k})$ for some $c,d \in \mathbb Z\}$ (Set of all polynomials of controlled degree and coefficient size w.r.t $k$)
$(2)$ $R[k] = \{(x,n) \in \mathbb Z^{2}: x$ ~ $O(\log^{r} {k}), n$ ~ $O(k^{s})$ for some $r,s \in \mathbb Z\}$ (Set of all pairs of integer points of prescribed size w.r.t $k$)
$(3)$ $S(k,P(x)) = \{(x,n) \in R[k] : P(x) = n!\}$ (Set of pairs of points of controlled size w.r.t $k$ satisfying a given equation)
$(4)$ $S(k) = \{S(k,P(x)):P(x) \in P[k]\}$ (set of pairs of points of controlled size w.r.t $k$ satisfying $P(x)=n!$ where $P(x)$ is of controlled size w.r.t $k$)
$(5)$ $S = \{S(k):k \in \mathbb Z\}$. (Set of $S(k)$)
$(6)$ $N = \{n \in \mathbb Z: \exists (x,n) \in S\}$. (Set of all integers which has a pair point $x$ so that $(x,n)$ form a controlled size pair solution to $P(x)=n!$ where $P(x)$ is of controlled size for some value of $k$)
$(7)$ $\epsilon$ = density of $N$ in $\mathbb Z$.
Conjecture: $\epsilon = 0$.
$\epsilon > 0$ implies for most $m \in \mathbb Z$, $\exists n$ close to $m$ such that $n!$ can be efficiently found since a reasonable size $k$ and $P(x)$ exists to compute $n!$. So $m!$ can be found using a constant number of multiplies given $n!$.
Though $\epsilon > 0$ implies $m!$ has an efficient algorithm for most $m \in \mathbb Z$ since $\exists n \in N$ close to the given $m$, it does not mean the algorithm is known since none of $n$, $k$, $P(x)$ and $x$ are explicit.
Are my arguments correct?
