# Brocard's problem

 Unsolved problem in mathematics: Does ${\displaystyle n!+1=m^{2}}$ have integer solutions other than ${\displaystyle n=4,5,7}$? (more unsolved problems in mathematics)

Brocard's problem is a problem in mathematics that asks to find integer values of n and m for which

${\displaystyle n!+1=m^{2},}$

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.

## 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. Matson (2017) has recently claimed to have extended this by 3 orders of magnitude to one trillion.

## Variants of the problem

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

${\displaystyle n!+A=k^{2}}$

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

${\displaystyle n!=P(x)}$

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

Cushing & Pascoe (2016) further varied the problem and have shown that it would follow from the abc conjecture that

${\displaystyle n!+K=m,}$

has only finitely many solutions, where K is an integer and ${\displaystyle m=a^{2}b^{3}}$ is a powerful number.