# 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Â¡th, “Über diophantische Gleichungen der Form $n!=x^{p}\pm y^{p}$ und $n!\pm m!=x^{p}$Acta Szeged. 8 (1937): 241 - 255
Title Brocard’s problem BrocardsProblem 2013-03-22 18:09:59 2013-03-22 18:09:59 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Definition msc 11A25