Woodall number

In number theory, a Woodall number (W_n) is any natural number of the form

: $W_n = n \cdot 2^n - 1$

for some natural number n. The first few Woodall numbers are:

:1, 7, 23, 63, 159, 383, 895, … {{OEIS|id=A003261}}.

History

Woodall numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917,{{citation

| last1 = Cunningham | first1 = A. J. C | author1-link = Allan Joseph Champneys Cunningham

| last2 = Woodall | first2 = H. J. | author2-link = H. J. Woodall

| journal = Messenger of Mathematics

| pages = 1–38

| title = Factorisation of $Q = (2^q \mp q)$ and $(q \cdot {2^q} \mp 1)$

| volume = 47

| year = 1917}}. inspired by James Cullen's earlier study of the similarly defined Cullen numbers.

Woodall primes

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

Woodall numbers that are also prime numbers are called Woodall primes; the first few exponents n for which the corresponding Woodall numbers W_n 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.{{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}} In October 1995, Wilfred Keller published a paper discussing several new Cullen primes and the efforts made to 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^{n + a} + b}}, where a and b are integers, and in particular, that almost all Woodall numbers are composite.{{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}} It is an open problem whether there are infinitely many Woodall primes. {{As of|2018|10}}, the largest known Woodall prime is 17016602 × 2^17016602 − 1.{{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}} It has 5,122,515 digits and was found by Diego Bertolotti in March 2018 in the distributed computing project PrimeGrid.{{Citation|author=PrimeGrid|author-link=PrimeGrid|title=Announcement of 17016602*2^17016602 - 1|url=http://www.primegrid.com/download/WOO-17016602.pdf|access-date=April 1, 2018}}

Restrictions

Starting with W₄ = 63 and W₅ = 159, every sixth Woodall number is divisible by 3; thus, in order for W_n to be prime, the index n cannot be congruent to 4 or 5 (modulo 6). Also, for a positive integer m, the Woodall number W_2^m may be prime only if 2^m + m is prime. As of January 2019, the only known primes that are both Woodall primes and Mersenne primes are W₂ = M₃ = 7, and W₅₁₂ = M₅₂₁.

Divisibility properties

Like Cullen numbers, Woodall numbers have many divisibility properties. For example, if p is a prime number, then p divides

:W_{(p + 1) / 2} if the Jacobi symbol $\left(\frac{2}{p}\right)$ is +1 and

:W_{(3p − 1) / 2} if the Jacobi symbol $\left(\frac{2}{p}\right)$ is −1.{{Citation needed|date=December 2011}}

Generalization

A generalized Woodall number base b 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 Woodall prime.

The smallest value of n such that n × bⁿ − 1 is prime for b = 1, 2, 3, ... are[http://harvey563.tripod.com/GWlist.txt List of generalized Woodall primes base 3 to 10000]

:3, 2, 1, 1, 8, 1, 2, 1, 10, 2, 2, 1, 2, 1, 2, 167, 2, 1, 12, 1, 2, 2, 29028, 1, 2, 3, 10, 2, 26850, 1, 8, 1, 42, 2, 6, 2, 24, 1, 2, 3, 2, 1, 2, 1, 2, 2, 140, 1, 2, 2, 22, 2, 8, 1, 2064, 2, 468, 6, 2, 1, 362, 1, 2, 2, 6, 3, 26, 1, 2, 3, 20, 1, 2, 1, 28, 2, 38, 5, 3024, 1, 2, 81, 858, 1, 2, 3, 2, 8, 60, 1, 2, 2, 10, 5, 2, 7, 182, 1, 17782, 3, ... {{OEIS|id=A240235}}

{{As of|2021|11}}, the largest known generalized Woodall prime with base greater than 2 is 2740879 × 32^2740879 − 1.{{cite web |title=The Top Twenty: Generalized Woodall |url=https://primes.utm.edu/top20/page.php?id=45 |website=primes.utm.edu |access-date=20 November 2021}}

References

External links

Chris Caldwell, [http://primes.utm.edu/glossary/page.php?sort=WoodallNumber The Prime Glossary: Woodall number], and [http://primes.utm.edu/top20/page.php?id=7 The Top Twenty: Woodall], and [http://primes.utm.edu/top20/page.php?id=45 The Top Twenty: Generalized Woodall], at The Prime Pages.
{{MathWorld|urlname=WoodallNumber|title=Woodall number}}
Steven Harvey, [http://harvey563.tripod.com/GeneralizedWoodallPrimes.txt List of Generalized Woodall primes].
Paul Leyland, [https://web.archive.org/web/20120204131629/http://www.leyland.vispa.com/numth/factorization/cullen_woodall/gcw.htm Generalized Cullen and Woodall Numbers]

Category:Integer sequences

Category:Unsolved problems in number theory

Category:Classes of prime numbers