Solutions of $p!q! = r!$

number-theory factorial diophantine-equations

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.


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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