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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.

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  • $\begingroup$ (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.) $\endgroup$ Commented May 26, 2012 at 23:01
  • $\begingroup$ 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 $r$. $\endgroup$
    – TMM
    Commented May 26, 2012 at 23:48
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    $\begingroup$ I think $6!7!=10!$ is the only known nontrivial solution, and it's conjectured that there aren't any others. $\endgroup$
    – MJD
    Commented May 27, 2012 at 0:03
  • $\begingroup$ @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. $\endgroup$ Commented May 27, 2012 at 0:03
  • $\begingroup$ @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. $\endgroup$ Commented May 27, 2012 at 8:01

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The only citation of the Luca paper found by MathSciNet:

Bhat, K. G.; Ramachandra, K.: A remark on factorials that are products of factorials. (Russian. Russian summary) Mat. Zametki 88 (2010), no. 3, 350–354; translation in Math. Notes 88 (2010), no. 3–4, 317–320

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