Problem #398: Are the only solutions to\[n!=x^2-1\]when $n=4,5,7$?

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Progress

The Brocard-Ramanujan conjecture. Erdős and Graham describe this as an old conjecture, and write it 'is almost certainly true but it is intractable at present'.

Overholt [Ov93] has shown that this has only finitely many solutions assuming a weak form of the ABC conjecture.

There are no other solutions below $10^9$ (see the OEIS page).