Statement:

Does the equation $$n!+1=m^2$$ have integer solutions other than \(n=4,5,7\)?

Author	Date Published	Prize	Paid By
Henri Brocard & Srinivasa Ramanujan (Independently)	1876, 1885 & 1913 respectively

Partial Results:

Integer pairs that solve Brocard's problem are known as Brown numbers. There are only three known pairs: \( (4,5),(5,11),(7,71)\)

Related Problems:

References:

Berndt, Bruce C.; Galway, William F. (2000), "The Brocard–Ramanujan diophantine equation n! + 1 = m2" (PDF), The Ramanujan Journal, 4: 41–42, doi:10.1023/A:1009873805276, S2CID 119711158.
Weisstein, Eric W. "Brocard's Problem". MathWorld.
https://mathworld.wolfram.com/BrownNumbers.html

Keywords:

Brocard, Brown, Ramanujan