Editing Woodall number (section)

==Woodall primes==

{{unsolved|mathematics|Are there infinitely many Woodall primes?}}

Woodall numbers that are also [[prime number]]s are called '''Woodall primes'''; the first few exponents ''n'' for which the corresponding Woodall numbers ''W''<sub>''n''</sub> are prime are 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, ... {{OEIS|id=A002234}}; the Woodall primes themselves begin with 7, 23, 383, 32212254719, ... {{OEIS|id=A050918}}.

In 1976 [[Christopher Hooley]] showed that [[almost all]] Cullen numbers are [[composite number|composite]].<ref name="EPSW94">{{cite book|last1=Everest|first1=Graham|title=Recurrence sequences|last2=van der Poorten|first2=Alf|last3=Shparlinski|first3=Igor|last4=Ward|first4=Thomas|publisher=[[American Mathematical Society]]|year=2003|isbn=0-8218-3387-1|series=Mathematical Surveys and Monographs|volume=104|location=[[Providence, RI]]|page=94|zbl=1033.11006|author2-link=Alfred van der Poorten}}</ref> In October 1995, Wilfred Keller published a paper discussing several new Cullen primes and the efforts made to [[Prime factorisation|factorise]] other Cullen and Woodall numbers. Included in that paper is a personal communication to Keller from [[Hiromi Suyama]], asserting that Hooley's method can be reformulated to show that it works for any sequence of numbers {{math|''n'' · 2<sup>''n'' + ''a''</sup> + ''b''}}, where ''a'' and ''b'' are [[integer]]s, and in particular, that almost all Woodall numbers are composite.<ref>{{Cite journal|last1=Keller|first1=Wilfrid|date=January 1995|title=New Cullen primes|journal=[[Mathematics of Computation]]|volume=64|issue=212|pages=1739|language=en|doi=10.1090/S0025-5718-1995-1308456-3|issn=0025-5718|doi-access=free}} {{Cite web|last1=Keller|first1=Wilfrid|date=December 2013|title=Wilfrid Keller|website=www.fermatsearch.org|location=Hamburg|language=en|url=http://www.fermatsearch.org/history/WKeller.html|access-date=October 1, 2020|url-status=live|archive-url=https://web.archive.org/web/20200228175855/http://www.fermatsearch.org/history/WKeller.html|archive-date=February 28, 2020}}</ref> It is an [[List of unsolved problems in mathematics#Prime numbers|open problem]] whether there are infinitely many Woodall primes. {{As of|2018|10}}, the largest known Woodall prime is 17016602&nbsp;×&nbsp;2<sup>17016602</sup>&nbsp;−&thinsp;1.<ref>{{Citation|title=The Prime Database: 8508301*2^17016603-1|url=http://primes.utm.edu/primes/page.php?id=124539|work=Chris Caldwell's The Largest Known Primes Database|access-date=March 24, 2018}}</ref> It has 5,122,515 digits and was found by Diego Bertolotti in March 2018 in the [[distributed computing]] project [[PrimeGrid]].<ref>{{Citation|author=PrimeGrid|author-link=PrimeGrid|title=Announcement of 17016602*2^17016602&nbsp;-&nbsp;1|url=http://www.primegrid.com/download/WOO-17016602.pdf|access-date=April 1, 2018}}</ref>