Brocard's problem is a problem in

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

that asks to find

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

values of

n

and

m

for which

n!+1 = m^2,

where

n!

is the

factorial In mathematics, the factorial of a non-negative denoted is the 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 (n-1) \times (n-2) \t ...

. It 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 point ...

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

Srinivasa Ramanujan Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis ...

Brown numbers

Pairs of the numbers

(n,m)

that solve Brocard's problem were named Brown numbers by

Clifford A. Pickover Clifford Alan Pickover (born August 15, 1957) is an American author, editor, and columnist in the fields of science, mathematics, science fiction, innovation, and creativity. For many years, he was employed at the IBM Thomas J. Watson Research ...

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 Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...

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 The ''abc'' conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. It is stated in terms of three positive integers ''a'', ''b'' ...

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 Brady John Haran (born 18 June 1976) is an Australian-British independent filmmaker and video journalist who produces educational videos and documentary films for his YouTube channels, the most notable being ''Periodic Videos'' and ''Number ...

, 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

Brown numbers

Connection to the abc conjecture

References

Further reading

External links