Brocard's problem

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

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

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)

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 2019, 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.

External links

• Weisstein, Eric W. "Brocard's Problem". MathWorld.
• Weisstein, Eric W. "Brown Numbers". MathWorld.
• Copeland, Ed. "Brown Numbers". Numberphile. Brady Haran. Archived from the original on 2014-11-09. Retrieved 2013-04-06.