Brocard's problem

Brocard's problem asks to find integer values of "n" for which

:$n!+1\; =\; m^2,$

where n! is the factorial. It was posed by H. Brocard in a pair of articles in 1876 and 1885, and independently in 1913 by 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. Most recently harvtxt|Berndt|Galway|2000 performed calculations for "n" up to 10⁹ and found no further solutions.

Variants of the problem

Dabrowski (1996) has shown that it would follow from the abc conjecture that

:$n!+A\; =\; k^2$

has only finitely many solutions, for any given integer "A".

References

*citation
last1 = Berndt | first1 = Bruce C.
last2 = Galway | first2 = William F.
title = The Brocard–Ramanujan diophantine equation "n"! + 1 = "m"²
journal = The Ramanujan Journal
url = http://www.math.uiuc.edu/~berndt/articles/galway.pdf
volume = 4 | year = 2000 | pages = 41–42.

*citation
last = Brocard | first = H.
title = Question 166
journal = Nouv. Corres. Math. | volume = 2 | pages = 287 | year = 1876.

*citation
last = Brocard | first = H.
title = Question 1532
journal = Nouv. Ann. Math. | volume = 4 | pages = 391 | year = 1885.

*citation
last = Dabrowski | first = A.
title = On the Diophantine Equation "x"! + "A" = "y"²
journal = Nieuw Arch. Wisk.
volume = 14
pages = 321–324
year = 1996.

*citation
last = Guy | first = R. K. | authorlink = Richard K. Guy
contribution = D25: Equations Involving Factorial
title = Unsolved Problems in Number Theory
edition = 2nd
location = New York | publisher = Springer-Verlag | pages = 193–194 | year = 1994.

External links

*mathworld|title=Brocard's Problem|urlname=BrocardsProblem
*mathworld|title=Brown Numbers|urlname=BrownNumbers