Are there any positive integer solutions $(n,x)$ to the equation $(x)(x+1)=n!$ except $(2,1)$ and $(3,2)$?
If not (as I suspect is the case), how do you prove that?
In general, is there a way to approach the Diophantine equation $n!=P(x)$ where $P$ is a polynomial?
This question was discussed on mathoverflow, and the question is open.