Brocard's problem is a problem in

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

that seeks

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

values of

n

such that

n!+1

is a perfect square, where

n!

is the

factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...

. Only three values of

n

are known — 4, 5, 7 — and it is not known whether there are any more. More formally, it seeks pairs of integers

n

and

m

such that

n!+1 = m^2.

The problem was posed by

Henri Brocard Pierre René Jean Baptiste Henri Brocard (; 12 May 1845 – 16 January 1922) was a French meteorologist and mathematician, in particular a geometer. His best-known achievement is the invention and discovery of the properties of the Brocard p ...

in a pair of articles in 1876 and 1885, and independently in 1913 by

Srinivasa Ramanujan Srinivasa Ramanujan Aiyangar (22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial con ...

Brown numbers

Pairs of the numbers

(n,m)

that solve Brocard's problem were named Brown numbers by Clifford A. Pickover in his 1995 book ''Keys to Infinity'', after learning of the problem from Kevin S. Brown. As of October 2022, there are only three known pairs of Brown numbers: based on the equalities Paul Erdős conjectured that no other solutions exist. Computational searches up to one quadrillion have found no further solutions.

Connection to the abc conjecture

It would follow from the

abc conjecture ABC are the first three letters of the Latin script. ABC or abc may also refer to: Arts, entertainment and media Broadcasting * Aliw Broadcasting Corporation, Philippine broadcast company * American Broadcasting Company, a commercial American ...

that there are only finitely many Brown numbers. More generally, it would also follow from the abc conjecture that

n!+A = k^2

has only finitely many solutions, for any given integer

A

, and that

n! = P(x)

has only finitely many integer solutions, for any given polynomial

P(x)

of degree at least 2 with integer coefficients.

References

External links

* * {{citation, last=Copeland, first=Ed, title=Brown Numbers, url=http://www.numberphile.com/videos/brown_numbers.html, work=Numberphile, publisher= Brady Haran, access-date=2013-04-06, archive-url=https://web.archive.org/web/20141109235028/http://www.numberphile.com/videos/brown_numbers.html, archive-date=2014-11-09, url-status=dead Diophantine equations Srinivasa Ramanujan Unsolved problems in number theory Factorial and binomial topics Abc conjecture

Brown numbers

Connection to the abc conjecture

References

Further reading

External links