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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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I think you mean $p=n$, $q=n!-1$ and $r=n!$. –  Thomas Andrews May 26 '12 at 22:59
    
(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.) –  Thomas Andrews May 26 '12 at 23:01
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Oops. I was a bit too generous with the "!". –  Rick Decker May 26 '12 at 23:26
    
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 May 26 '12 at 23:48
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I think $6!7!=10!$ is the only known nontrivial solution, and it's conjectured that there aren't any others. –  MJD May 27 '12 at 0:03
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up vote 3 down vote accepted

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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