I am trying to collect informations concerning the set $$\mathcal{A}=\left\{n\in\mathbb{N} \mid (\exists k,l\in\{2,3,\dots,n-2\})(n!=k!l!)\right\}.$$ It seems not much is known about the set $\mathcal{A}$. Below, I mention all what I have. Any other result relating properties of $\mathcal{A}$ will be appreciated.

First, it is not known (to my best knowledge), if $\mathcal{A}$ is finite or not. Note that the range for factors $k$ and $l$ excludes values $1,n$, and also $n-1$. Clearly, I want to avoid trivial identities of the form $n!=1!n!$. Less trivially, nevertheless it not an interesting case if one factor equlas $n-1$ since for any integer $n$ which is a factorial of another integer, say $n=m!$, we have $n!=m!(n-1)!$. For example, $6!=3!5!$, but $6\notin\mathcal{A}$.

The least number and the only one I know belonging to $\mathcal{A}$ is 10 since $$10!=6!7!.$$

I can prove (and it is not hard at all) that if $n\in\mathcal{A}$ and $n!=k!l!$, then $k+l>n+1$.

Mathematicians studied a more general set allowing decomposition of $n!$ into a product of more then only two factorials (hence $\mathcal{A}$ is a subset). In $\S$B23 (page 80) of
R. K. Guy, *Unsolved Problems in Number Theory*, 2nd ed., New York, Springer-Verlag, 1994, one finds the following:

With the aid of computer, J.Shallit and M. Easter showed that, between numbers $1,2,\dots,18160$, only the number 10 belongs to $\mathcal{A}$. This computation comes from early 90's and, as I indicated above, it treats even more general problem. So I thing that at least this result could be extended significantly with today's computers. I do not know the algorithm used by J.Shallit and M. Easter but certainly the main obstacle is that one has to check a lot of equalities between huge numbers. This could be simplified by taking a logarithm of both sides in $n!=k!l!$, for instance.

Next, it has been proved by Paul Erdős that if $$\lim_{n\to\infty}\frac{P(n(n+1))}{\ln n}=\infty,$$ where $P(n)$ is the largest prime factor of $n$, then $\mathcal{A}$ has to be finite.

Finally, by personal communications with some colleges (mathematicians, though not number theorists) I encountered also a curios opinion that $\mathcal{A}$ contains **only** number 10.