Brocard's problem

Not to be confused with Brocard's conjecture.

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

Unsolved problem in mathematics:

Do integers, n,m, exist such that n!+1=m² other than n=4,5,7?
(more unsolved problems in mathematics)

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.

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 .
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 .
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.

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