Brocard's problem asks to find integer values of n and m for which n! + 1 = m2, where n is the factorial.
Put another way:
Does the equation n!+1 = m2 have integer solutions other than 4, 5, 7?
It was first proposed by French mathematician Henri Brocard 1876.
Brocard found just three solutions : the number pairs [4,5] [5,11] and [7,71].
As yet, no other solutions have been found, even with very extensive computational searches.
It's currently not known if there are or are not any other solutions - and no proof exists either way.
See : Wikipedia
Further technical investigations (2023) : The diagonalization method and Brocard's problem, arXiv, math,1803.09155
Also see : Brocard's conjectureplugin-autotooltip__plain plugin-autotooltip_bigBrocard's conjecture
Brocard's conjecture asserts that there are at least four prime numbers between (pn)2 and (pn+1)2, where pn is the nth prime number, for every n ≥ 2.
It was first suggested by French mathematician Henri Brocard in the late 19th century.