# Solutions of $p!q! = r!$

The title says it all, more or less. Obviously, there are infinitely many "trivial" integral solutions of the form $p=n, q=(n!-1), r= n!$. How many non-trivial solutions are there?

I came across this about ten years ago; as far as I can tell, it hasn't appeared here before, so I thought that it might be of interest. I'm actually most interested in finding whether there was any progress made since Florian Luca's 2007 article.

• (It's weird that Luca's extract at the beginning calls this a Diophantine equation. That's not any meaning of "Diophantine" than any I've seen.) Commented May 26, 2012 at 23:01
• For what it's worth, if $p \leq q < r$ then there obviously cannot be any primes in $\{q+1, \ldots, r\}$. So $q$ has to be 'close' to $r$.
– TMM
Commented May 26, 2012 at 23:48
• I think $6!7!=10!$ is the only known nontrivial solution, and it's conjectured that there aren't any others.
– MJD
Commented May 27, 2012 at 0:03
• @TMM Right. That's why it's likely that there will only be a finite number of nontrivial solutions (the first one, actually, is fairly small). Also, I agree that MO would be an appropriate place to ask. Commented May 27, 2012 at 0:03
• @ThomasAndrews, it is not uncommon for number theorists to refer to just about any old equation as a Diophantine equation, if their interest lies in integral solutions to the equation. Commented May 27, 2012 at 8:01