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Brocard's problem (Definition)

Brocard's problem, first posed by Henri Brocard in 1876, asks for factorials that are one less than a square, that is, solutions to the equation $ n! + 1 = m^2$. Only three solutions are known: $ 4! + 1 = 5^2$, $ 5! + 1 = 11^2$ and $ 7! + 1 = 71^2$. Srinivasa Ramanujan also pondered the problem, in 1913. Erdős believed that there are no other solutions, and no more have been found for $ n$ up to $ 10^9$.

Bibliography

1
P. Erdős, & R. Obláth, ``Über diophantische Gleichungen der Form $ n! = x^p \pm y^p$ und $ n! \pm m! = x^p$'' Acta Szeged. 8 (1937): 241 - 255



"Brocard's problem" is owned by PrimeFan.
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table of differences between $\lceil \sqrt{n!} \rceil^2$ and $n!$ for $0 < n < 26$ (Data Structure) by PrimeFan
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Cross-references: Srinivasa Ramanujan, equation, solutions, square, factorials
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This is version 1 of Brocard's problem, born on 2008-06-29.
Object id is 10727, canonical name is BrocardsProblem.
Accessed 171 times total.

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AMS MSC: 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)
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