Are there infinitely many nontrivial integer solutions $(a,b,c)$ of
$$\Gamma(a)\Gamma(b)=\Gamma(c)\hspace{10mm}?$$
This seems like it may have been asked before but I didn't find an earlier question (yet). I checked for about $c < 200$ and $b-a < 10$ and found only $(a,b,c)=(4,6,7),(7,8,11).$
Note that $(d+1, d! , d!+1)$ is a solution for all $d$:
$$\Gamma(d+1)\Gamma(d!) = d!(d!-1)! = (d!)! = \Gamma(d!+1)$$
However, this might be considered as a trivial solution. But since you included $(4,6,7)$, which is this solution with $d=3$, I thought it was worth pointing out.
I think the really trivial solutions are $(1,d,d)$ and $(2,d,d)$.
A bit more advanced stuff: Under the abc conjecture, there are only finitely many other solutions than those I mentioned above. Probably, the only one is $(7,8,11)$, which corresponds with $6!7!=10!$. See also: Solutions of $p!q! = r!$