# Wikenigma - an Encyclopedia of Unknowns Wikenigma - an Encyclopedia of the Unknown

# Brocard's problem

Brocard's problem asks to find integer values of *n* and *m* for which* n*! + 1 = *m*^{2}, where *n* is the factorial.

Put another way:

Does the equation n!+1 = m^{2} 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 â‰¥ 2.

It was first suggested by French mathematician Henri Brocard in the late 19th century.

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