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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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    @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.2012-05-27

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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