Brocard's problem

{{unsolved|mathematics|2=Does $n!+1=m^2$ have integer solutions other than $n=4,5,7$ ?}}

Brocard's problem is a problem in mathematics that seeks integer values of $n$ such that $n!+1$ is a perfect square, where $n!$ is the factorial. 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 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 $(n,m)$ 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.{{r|pickover}} As of October 2022, there are only three known pairs of Brown numbers:

based on the equalities

{{bi|left=1.6|4! + 1 {{=}} 5² {{=}} 25,}}

{{bi|left=1.6|5! + 1 {{=}} 11² {{=}} 121, and}}

{{bi|left=1.6|7! + 1 {{=}} 71² {{=}} 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

$n!+A = k^2$

has only finitely many solutions, for any given integer $A$ ,{{r|dabrowski}} 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.{{r|luca}}

References

{{reflist|refs=

{{citation

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| 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 {{=}} m²}}

| 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

}}

{{citation

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| title = Question 166

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{{citation

| last = Brocard | first = H.

| title = Question 1532

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{{citation

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| mr = 1430045

| pages = 321–324

| title = On the Diophantine equation {{math|x! + A {{=}} y²}}

| volume = 14

| year = 1996}}

{{citation

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| last2 = Glickman | first2 = Jacob

| title = C++ Brocard GitHub Repository

| url = https://github.com/jhg023/brocard

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{{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

{{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}}

{{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}}

{{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 {{=}} m²}}

| volume = 25

| year = 1993}}

{{citation

| last = Pickover | first = Clifford A. | author-link = Clifford A. Pickover

| page = 170

| publisher = John Wiley & Sons

| title = Keys to Infinity

| year = 1995}}

{{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}}

}}

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

Brocard's problem

Brown numbers

Connection to the abc conjecture

References

Further reading

External links