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Brocard's problem

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 โ‰ฅ 2.

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


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