On the factorial equations $A! B! =C!$ and $A!B!C! = D!$ I was playing around with hypergeometric probabilities when I wound myself calculating the binomial coefficient $\binom{10}{3}$. I used the definition, and calculating in my head, I simplified to this expression before actually calculating anything
$$
\frac {8\cdot9\cdot10}{2\cdot3} = 120
$$
And then it hit me that $8\cdot9\cdot10 = 6!$ and I started thinking about something I feel like calling generalized factorials, which is just the product of a number of successive naturals, like this
$$
a!b = \prod_{n=b}^an = \frac{a!}{(b-1)!},\quad a, b \in \mathbb{Z}^+, \quad a\ge b
$$
so that $a! = a!1$ (the notation was invented just now, and inspired by the $nCr$-notation for binomial coefficients). Now, apart from the trivial examples $(n!)!(n!) = n!$ and $a!1 = a!2 = a!$, when is the generalized factorial a factorial number? When is it the product of two (non-trivial) factorial numbers? As seen above, $10!8$ is both.

Consider the general equation $$a_1!a_2!\cdots a_n! = b!.$$
People like Paul Erdős worked on this kind of equations. So, they are rather serious problems. Even for the simplest case where $n=2$, namely the equation $A!B!=C!$, we don't know whether there is any other non-trivial solution other than $10!=7!6!$. 
Erdős proved that in the equation $A!B!=C!$ if we assume $1< A \leq B$, then for large enough values of $C$, the difference of $B$ and $C$ does not exceed $5 \log \log n$. It is clear that $5 \log \log n$ grows to infinity as $n$ increases. But in fact, this is a very slowly growing function. 
For instance, the first $n$ for which $5 \log \log n = 50$ has more than $9566$ digits. Caldwell proved (1994) that the only non-trivial solution with $C < 10^6$ is $10!=7!6!$. Florian Luca proved in this paper (2007) that considering the $abc$  hypothesis, we have $C-B=1$ for large enough $n$. Read also a remark (2009) on this paper if you're interested. 
Suppose that $P(m)$ denotes the largest prime factor of $m$. Erdős also showed in one of his many papers that the assertion
$$
P(n(n+1)) > 4 \log n
$$
would imply that the general equation $(1)$ has finitely many non-trivial solutions. 
The last paragraph is taken from a recent work of Hajdua, Pappa, and Szakács, who prove proved in this paper (2018) that writing $k=B-A$ for all non-trivial solutions of the equation $A!B!=C!$ different from $(A, B, C) =(6, 7, 10)$, we have $C<5k$. Further, if $k \leq 10^6$, then the only non-trivial solution is given by $(A, B, C) =(6, 7, 10)$. I presented their paper in our MathSciNet seminar at UBC and you can find the slides here. 
More references:


*

*Discussion on ArtofProblemSolving

*Luca, Florian. "The Diophantine equation $P (x)= n!$ and a result of M. Overholt." Glasnik matematički 37.2 (2002): 269-273.

*Berend, Daniel, and Jørgen Harmse. "On polynomial-factorial diophantine equations." Transactions of the American Mathematical Society 358.4 (2006): 1741-1779.

