====== Brocard's problem ======

Brocard's problem asks to find integer values of //n// and //m// for which// n//! + 1 = //m//<sup>2</sup>, where //n// is the factorial.

Put another way:

Does the equation n!+1 = m<sup>2</sup> 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 : [[https://en.wikipedia.org/wiki/Brocard%27s_problem|Wikipedia]]

Further technical investigations (2023) : [[https://arxiv.org/pdf/1803.09155.pdf|The diagonalization method and Brocard's problem]], arXiv, math,1803.09155

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Also see : [[content:mathematics:brocards_conjecture]]