View source for Brocard's problem

{{Short description|In mathematics, when is n!+1 a square}}
{{distinguish|Brocard's conjecture}}
{{unsolved|mathematics|2=Does <math>n!+1=m^2</math> have integer solutions other than <math>n=4,5,7</math>?}}
'''Brocard's problem''' is a problem in [[mathematics]] that seeks [[integer]] values of <math>n</math> such that <math>n!+1</math> is a perfect square, where <math>n!</math> is the [[factorial]]. Only three values of <math>n</math> are known &mdash; 4, 5, 7 &mdash; and it is not known whether there are any more.

More formally, it seeks pairs of integers <math>n</math> and <math>m</math> such that<math display="block">n!+1 = m^2.</math>The problem was posed by [[Henri Brocard]] in a pair of articles in 1876 and 1885,{{r|broc1|broc2}} and independently in 1913 by [[Srinivasa Ramanujan]].{{r|ramanujan}}

==Brown numbers==
Pairs of the numbers <math>(n,m)</math> 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.{{r|pickover}} As of October 2022, there are only three known pairs of Brown numbers:
{{bi|left=1.6|(4,5), (5,11), and (7,71),}}
based on the equalities
{{bi|left=1.6|4! + 1 {{=}} 5<sup>2</sup> {{=}} 25,}}
{{bi|left=1.6|5! + 1 {{=}} 11<sup>2</sup> {{=}} 121, and}}
{{bi|left=1.6|7! + 1 {{=}} 71<sup>2</sup> {{=}} 5041.}}
[[Paul Erdős]] conjectured that no other solutions exist.{{r|erdos}} Computational searches up to one quadrillion have found no further solutions.{{r|bergal|matson|epsgli}}

==Connection to the abc conjecture==
It would follow from the [[abc conjecture]] that there are only finitely many Brown numbers.{{r|overholt}}
More generally, it would also follow from the abc conjecture that
<math display=block>n!+A = k^2</math>
has only finitely many solutions, for any given integer <math>A</math>,{{r|dabrowski}} and that
<math display=block>n! = P(x)</math>
has only finitely many integer solutions, for any given polynomial <math>P(x)</math> of degree at least 2 with integer coefficients.{{r|luca}}

==References==
{{reflist|refs=

<ref name=bergal>{{citation
 | last1 = Berndt | first1 = Bruce C.
 | last2 = Galway | first2 = William F.
 | doi = 10.1023/A:1009873805276
 | issue = 1
 | journal = Ramanujan Journal
 | mr = 1754629
 | pages = 41–42
 | title = On the Brocard–Ramanujan Diophantine equation {{math|''n''! + 1 {{=}} ''m''<sup>2</sup>}}
 | url = https://www.math.uiuc.edu/~berndt/articles/galway.pdf
 |url-status=dead
 | archive-url = https://web.archive.org/web/20170703115126/https://www.math.uiuc.edu/~berndt/articles/galway.pdf
 | archive-date = 2017-07-03
 | volume = 4
 | year = 2000| s2cid = 119711158
 }}</ref>

<ref name=broc1>{{citation
 | last = Brocard | first = H.
 | title = Question 166
 | journal = Nouv. Corres. Math. | volume = 2 | pages = 287 | year = 1876}}</ref>

<ref name=broc2>{{citation
 | last = Brocard | first = H.
 | title = Question 1532
 | journal = Nouv. Ann. Math. | volume = 4 | pages = 391 | year = 1885}}</ref>

<ref name=dabrowski>{{citation
 | last = Dąbrowski | first = Andrzej
 | issue = 3
 | journal = Nieuw Archief voor Wiskunde
 | mr = 1430045
 | pages = 321–324
 | title = On the Diophantine equation {{math|''x''! + A {{=}} ''y''<sup>2</sup>}}
 | volume = 14
 | year = 1996}}</ref>

<ref name=epsgli>{{citation
 | last1 = Epstein | first1 = Andrew
 | last2 = Glickman | first2 = Jacob
 | title = C++ Brocard GitHub Repository
 | url = https://github.com/jhg023/brocard
 | year = 2020}}</ref>

<ref name=erdos>{{citation
 | last = Erdős | first = Paul | author-link = Paul Erdős
 | editor1-last = Chabauty | editor1-first = C.
 | editor2-last = Chatelet | editor2-first = A.
 | editor3-last = Chatelet | editor3-first = F.
 | editor4-last = Descombes | editor4-first = R.
 | editor5-last = Pisot | editor5-first = C.
 | editor6-last = Poitou | editor6-first = G.
 | contribution = Quelques problèmes de la théorie des nombres
 | contribution-url = https://users.renyi.hu/~p_erdos/1963-14.pdf
 | language = fr
 | pages = 81–135
 | publisher = University of Geneva
 | series = Monographies de l'Enseignement Mathématique
 | title = Introduction à la théorie des nombres
 | volume = 6
 | year = 1963}}; see problème 67, p. 129</ref>

<ref name=luca>{{citation
 | last = Luca | first = Florian
 | issue = 2
 | journal = Glasnik Matematički
 | mr = 1951531
 | pages = 269–273
 | title = The Diophantine equation {{math|''P''(''x'') {{=}} ''n''!}} and a result of M. Overholt
 | url = https://web.math.hr/glasnik/37.2/37(2)-04.pdf
 | volume = 37(57)
 | year = 2002}}</ref>
<ref name=matson>{{citation
 | last = Matson | first = Robert
 | title = Brocard's Problem 4th Solution Search Utilizing Quadratic Residues
 | url = http://unsolvedproblems.org/S99.pdf
 | journal = Unsolved Problems in Number Theory, Logic and Cryptography
 | year = 2017
 | access-date = 2017-05-07
 | archive-date = 2018-10-06
 | archive-url = https://web.archive.org/web/20181006100943/http://unsolvedproblems.org/S99.pdf
 | url-status = dead}}</ref>

<ref name=overholt>{{citation
 | last = Overholt | first = Marius
 | doi = 10.1112/blms/25.2.104
 | issue = 2
 | journal = The Bulletin of the London Mathematical Society
 | mr = 1204060
 | page = 104
 | title = The Diophantine equation {{math|''n''! + 1 {{=}} m<sup>2</sup>}}
 | volume = 25
 | year = 1993}}</ref>

<ref name=pickover>{{citation
 | last = Pickover | first = Clifford A. | author-link = Clifford A. Pickover
 | page = 170
 | publisher = John Wiley & Sons
 | title = Keys to Infinity
 | year = 1995}}</ref>

<ref name=ramanujan>{{citation
 | last = Ramanujan | first = Srinivasa
 | editor1-last = Hardy | editor1-first = G. H.
 | editor2-last = Aiyar | editor2-first = P. V. Seshu
 | editor3-last = Wilson | editor3-first = B. M.
 | contribution = Question 469
 | contribution-url = https://books.google.com/books?id=h1G2CgAAQBAJ&pg=PA327
 | isbn = 0-8218-2076-1
 | location = Providence, Rhode Island
 | mr = 2280843
 | page = 327
 | publisher = AMS Chelsea Publishing
 | title = Collected papers of Srinivasa Ramanujan
 | year = 2000}}</ref>

}}

==Further reading==
*{{citation
 | last = Guy | first = R. K. | authorlink = Richard K. Guy
 | contribution = D25: Equations involving factorial <math>n</math>
 | title = Unsolved Problems in Number Theory
 | edition = 3rd
 | location = New York | publisher = Springer-Verlag
 | pages = 301–302
 | year = 2004}}
*{{citation
 | last = Makki Naciri | first = Abderrahim
 | url = https://doi.org/10.1007/s11139-024-00960-0
 | issue = 2
 | journal = Ramanujan Journal
 | pages =  1791–1798
 | title = On the variant Q(n!)=P(x) of the Brocard-Ramanujan Diophantine equation.
 | volume = 65
 | year = 2024| doi = 10.1007/s11139-024-00960-0
 | url-access = subscription
 }}
==External links==
* {{mathworld2|title=Brocard's Problem|urlname=BrocardsProblem|title2=Brown Numbers|urlname2=BrownNumbers}}
* {{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}}

[[Category:Diophantine equations]]
[[Category:Srinivasa Ramanujan]]
[[Category:Unsolved problems in number theory]]
[[Category:Factorial and binomial topics]]
[[Category:Abc conjecture]]