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# Brocard’s problem

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 $10^{9}$.

# References

- 1 P. Erdős, & R. OblÃ¡th, “Über diophantische Gleichungen der Form $n!=x^{p}\pm y^{p}$ und $n!\pm m!=x^{p}$” Acta Szeged. 8 (1937): 241 - 255

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