Brocard's problem

Brocard's problem (not to be confused with Brocard's Conjecture) asks to find integer values of n for which

$n!+1 = m^2,$

where n! is the factorial. It was posed by Henri Brocard in a pair of articles in 1876 and 1885, and independently in 1913 by Srinivasa Ramanujan.

 Do integers, n,m, exist such that n!+1=m2 other than n=4,5,7?

Brown numbers

Pairs of the numbers (n, m) that solve Brocard's problem are called Brown numbers. There are only three known pairs of Brown numbers:

(4,5), (5,11), and (7,71).

Paul Erdős conjectured that no other solutions exist. Overholt (1993) showed that there are only finitely many solutions provided that the abc conjecture is true. Berndt & Galway (2000) performed calculations for n up to 109 and found no further solutions.

Variants of the problem

Dabrowski (1996) generalized Overholt's result by showing that it would follow from the abc conjecture that

$n!+A = k^2$

has only finitely many solutions, for any given integer A. This result was further generalized by Luca (2002), who showed (again assuming the abc conjecture) that the equation

$n! = P(x)$

has only finitely many integer solutions for a given polynomial P(x) of degree at least 2 with integer coefficients.