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Undid revision 631976601 by SBS6679D (talk). False and contradicts source

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In [[mathematics]], a '''Cullen number''' is a [[natural number]] of the form ''n'' · 2''n'' + 1 (written ''C''''n''). Cullen numbers were first studied by Fr. [[James Cullen (mathematician)|James Cullen]] in 1905. Cullen numbers are special cases of [[Proth number]]s.

== Properties ==

In 1976 [[Christopher Hooley]] showed that the [[natural density]] of positive integers <math>n \leq x</math> for which ''Cn'' is a prime is of the order ''o(x)'' for <math>x\to\infty</math>. In that sense, [[almost all]] Cullen numbers are [[composite number|composite]].<ref name=EPSW94>{{cite book | last1=Everest | first1=Graham | last2=van der Poorten | first2=Alf | author2-link=Alfred van der Poorten | last3=Shparlinski | first3=Igor | last4=Ward | first4=Thomas | title=Recurrence sequences | series=Mathematical Surveys and Monographs | volume=104 | location=[[Providence, RI]] | publisher=[[American Mathematical Society]] | year=2003 | isbn=0-8218-3387-1 | zbl=1033.11006 | page=94 }}</ref> Hooley's proof was reworked by [[Hiromi Suyama]] to show that it works for any sequence of numbers ''n'' · 2''n''+''a'' + ''b'' where ''a'' and ''b'' are integers, and in particular also for [[Woodall number]]s. The only known '''Cullen primes''' are those for ''n'' equal:
: 1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881 {{OEIS|id=A005849}}.
Still, it is conjectured that there are infinitely many Cullen primes.

{{As of|2009|08}}, the largest known Cullen prime is 6679881 × 26679881 + 1. It is a [[megaprime]] with 2,010,852 digits and was discovered by a [[PrimeGrid]] participant from Japan.<ref>{{Citation |url=http://primes.utm.edu/primes/page.php?id=89536 |title=The Prime Database: 6679881*2^6679881+1 |work=Chris Caldwell's The Largest Known Primes Database |accessdate=December 22, 2009 }}</ref>

A Cullen number ''Cn'' is divisible by ''p'' = 2''n'' − 1 if ''p'' is a [[prime number]] of the form 8''k'' - 3; furthermore, it follows from [[Fermat's little theorem]] that if ''p'' is an odd prime, then p divides ''C''''m''(''k'') for each ''m''(''k'') = (2''k'' − ''k'') 
 (''p'' − 1) − ''k'' (for ''k'' > 0). It has also been shown that the prime number ''p'' divides ''C''(''p'' + 1) / 2 when the [[Jacobi symbol]] (2 | ''p'') is −1, and that ''p'' divides ''C''(3''p'' − 1) / 2 when the Jacobi symbol (2 | ''p'') is +1.

It is unknown whether there exists a prime number ''p'' such that ''C''''p'' is also prime.

== Generalizations ==

Sometimes, a '''generalized Cullen number''' is defined to be a number of the form ''n'' · ''bn'' + 1, where ''n'' + 2 > ''b''; if a prime can be written in this form, it is then called a '''generalized Cullen prime'''. Woodall numbers are sometimes called '''Cullen numbers of the second kind'''.

{{As of|2012|02}}, the largest known generalized Cullen prime is 427194 × 113 427194 + 1. It has 877,069 digits and was discovered by a [[PrimeGrid]] participant from United States.<ref>{{Citation|url= http://primes.utm.edu/primes/page.php?id=104121 |title=The Prime Database: 427194 · 113^427194 + 1 |work=Chris Caldwell's The Largest Known Primes Database |accessdate=January 30, 2012 }}</ref>

== References ==
{{Reflist}}

==Further reading==
* {{Citation |last=Cullen |first=James |title=Question 15897 |journal=Educ. Times |month=December |year=1905 |page=534}}.
* {{Citation |first=Richard K. |last=Guy |authorlink=Richard K. Guy |title=Unsolved Problems in Number Theory |edition=3rd |publisher=[[Springer Verlag]] |location=New York |year=2004 |isbn=0-387-20860-7 | zbl=1058.11001 | at=Section B20 }}.
* {{Citation |last=Hooley |first=Christopher |authorlink=Christopher Hooley |title=Applications of sieve methods |publisher=[[Cambridge University Press]] |year=1976 |isbn=0-521-20915-3 |pages=115–119 | zbl=0327.10044 | series=Cambridge Tracts in Mathematics | volume=70 }}.
* {{Citation |first=Wilfrid |last=Keller |title=New Cullen Primes |journal=[[Mathematics of Computation]] |volume=64 |issue=212 |year=1995 |pages=1733–1741,S39–S46 |url=http://www.ams.org/mcom/1995-64-212/S0025-5718-1995-1308456-3/S0025-5718-1995-1308456-3.pdf | zbl=0851.11003 | issn=0025-5718 }}.

== External links ==
* Chris Caldwell, [http://primes.utm.edu/top20/page.php?id=6 The Top Twenty: Cullen primes] at The [[Prime Pages]].
* [http://primes.utm.edu/glossary/page.php?sort=Cullens The Prime Glossary: Cullen number] at The Prime Pages.
* {{MathWorld | urlname=CullenNumber | title=Cullen number}}
* [http://www.prothsearch.net/cullen.html Cullen prime: definition and status] (outdated), Cullen Prime Search is now hosted at [[PrimeGrid]]
* Paul Leyland, [http://www.leyland.vispa.com/numth/factorization/cullen_woodall/gcw.htm Generalized Cullen and Woodall Numbers]

{{Prime number classes|state=collapsed}}
{{Classes of natural numbers}}

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[[Category:Integer sequences]]
[[Category:Unsolved problems in mathematics]]

Deficient number - Revision history

en>PrimeHunter: Undid revision 631976601 by SBS6679D (talk). False and contradicts source

en>Maghnus: Undid revision 553274927 by 76.183.94.212 (talk)