The facts "standard heuristic arguments suggest that there are only finitely many solutions" and "extensive numerical search shows that there are no unknown small solutions" are strong evidence for there being no more solutions. (Apparently also the ABC-conjecture implies that there are only finitely many solutions.)

https://en.wikipedia.org/wiki/Brocard%27s_problem

Resolves once a proof is well-accepted among the mathematical community. If it's proven to be unprovable or not well-defined, resolves N/A.

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