# Brocard's problem

Unsolved problem in mathematics:

Does ${\displaystyle n!+1=m^{2}}$ have integer solutions other than ${\displaystyle n=4,5,7}$?

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. As of 2020, 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 extended this by three orders of magnitude to one trillion. Epstein & Glickman (2020) have recently extended this by three more orders of magnitude to one quadrillion.

## 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.

## 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.
• Brocard, H. (1876), "Question 166", Nouv. Corres. Math., 2: 287.
• Brocard, H. (1885), "Question 1532", Nouv. Ann. Math., 4: 391.
• Dabrowski, A. (1996), "On the Diophantine Equation x! + A = y2", Nieuw Arch. Wisk., 14: 321–324.
• Epstein, Andrew; Glickman, Jacob (2020), C++ Brocard GitHub Repository.
• Guy, R. K. (1994), "D25: Equations Involving Factorial", Unsolved Problems in Number Theory (2nd ed.), New York: Springer-Verlag, pp. 193–194, ISBN 0-387-90593-6.
• Luca, Florian (2002), "The diophantine equation P(x) = n! and a result of M. Overholt" (PDF), Glasnik Matematički, 37 (57): 269–273.
• Matson, Robert (2017), "Brocard's Problem 4th Solution Search Utilizing Quadratic Residues" (PDF), Unsolved Problems in Number Theory, Logic and Cryptography.
• Overholt, Marius (1993), "The diophantine equation n! + 1 = m2", Bull. London Math. Soc., 25 (2): 104, doi:10.1112/blms/25.2.104.