Woodall number: Difference between revisions
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{{Short description|Number of the form (n * 2^n) - 1}} |
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In [[number theory]], a '''Woodall number''' (W<sub>n</sub>) is any [[natural number]] of the form |
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{{pp-semi-indef|small=yes}} |
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In [[number theory]], a '''Woodall number''' (''W''<sub>''n''</sub>) is any [[natural number]] of the form |
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:<math>W_n = n \cdot 2^n - 1</math> |
:<math>W_n = n \cdot 2^n - 1</math> |
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| title = Factorisation of <math>Q = (2^q \mp q)</math> and <math>(q \cdot {2^q} \mp 1)</math> |
| title = Factorisation of <math>Q = (2^q \mp q)</math> and <math>(q \cdot {2^q} \mp 1)</math> |
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| volume = 47 |
| volume = 47 |
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| year = 1917}}.</ref> inspired by [[James Cullen (mathematician)|James Cullen]]'s earlier study of the similarly |
| year = 1917}}.</ref> inspired by [[James Cullen (mathematician)|James Cullen]]'s earlier study of the similarly defined [[Cullen number]]s. |
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==Woodall primes== |
==Woodall primes== |
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{{unsolved|mathematics|Are there infinitely many Woodall primes?}} |
{{unsolved|mathematics|Are there infinitely many Woodall primes?}} |
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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, |
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}}. |
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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 |
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 × 2<sup>17016602</sup> − 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 - 1|url=http://www.primegrid.com/download/WOO-17016602.pdf|access-date=April 1, 2018}}</ref> |
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==Restrictions== |
==Restrictions== |
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Starting with W<sub>4</sub> = 63 and W<sub>5</sub> = 159, every sixth Woodall number is divisible by 3; thus, in order for W<sub>n</sub> 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<sub>2<sup>m</sup></sub> may be prime only if 2<sup>m</sup> + m is prime. As of January 2019, the only known primes that are both Woodall primes and [[Mersenne primes]] are W<sub>2</sub> = M<sub>3</sub> = 7, and W<sub>512</sub> = M<sub>521</sub>. |
Starting with ''W''<sub>4</sub> = 63 and ''W''<sub>5</sub> = 159, every sixth Woodall number is [[divisible]] by 3; thus, in order for ''W''<sub>''n''</sub> to be prime, the index ''n'' cannot be [[modular arithmetic|congruent]] to 4 or 5 (modulo 6). Also, for a positive integer ''m'', the Woodall number ''W''<sub>2<sup>''m''</sup></sub> may be prime only if 2<sup>''m''</sup> + ''m'' is prime. As of January 2019, the only known primes that are both Woodall primes and [[Mersenne primes]] are ''W''<sub>2</sub> = ''M''<sub>3</sub> = 7, and ''W''<sub>512</sub> = ''M''<sub>521</sub>. |
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==Divisibility properties== |
==Divisibility properties== |
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Like Cullen numbers, Woodall numbers have many divisibility properties. For example, if ''p'' is a prime number, then ''p'' divides |
Like Cullen numbers, Woodall numbers have many divisibility properties. For example, if ''p'' is a prime number, then ''p'' divides |
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:''W''<sub>(''p'' |
:''W''<sub>(''p'' + 1) / 2</sub> if the [[Jacobi symbol]] <math>\left(\frac{2}{p}\right)</math> is +1 and |
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:''W''<sub>(3''p'' −& |
:''W''<sub>(3''p'' − 1) / 2</sub> if the Jacobi symbol <math>\left(\frac{2}{p}\right)</math> is −1.{{Citation needed|date=December 2011}} |
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==Generalization== |
==Generalization== |
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A '''generalized Woodall number base ''b''''' is defined to be a number of the form ''n'' × ''b''<sup>''n''</sup> − 1, where ''n'' + 2 > ''b''; if a prime can be written in this form, it is then called a '''generalized Woodall prime'''. |
A '''generalized Woodall number base ''b''''' is defined to be a number of the form ''n'' × ''b''<sup>''n''</sup> − 1, where ''n'' + 2 > ''b''; if a prime can be written in this form, it is then called a '''generalized Woodall prime'''. |
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The smallest value of ''n'' such that ''n'' × ''b''<sup>''n''</sup> − 1 is prime for ''b'' = 1, 2, 3, ... are<ref name="tripod">[http://harvey563.tripod.com/GWlist.txt List of generalized Woodall primes base 3 to 10000]</ref> |
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: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}} |
: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}} |
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{{As of|2021|11}}, the largest known generalized Woodall prime with base greater than 2 is 2740879 × 32<sup>2740879</sup> − 1.<ref>{{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}}</ref> |
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{|class="wikitable" |
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|''b'' |
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|numbers ''n'' such that ''n'' × ''b''<sup>''n''</sup> - 1 is prime (these ''n'' are checked up to 350000) |
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|[[OEIS]] sequence |
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|1 |
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|3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278, 282, 284, 294, ... (all primes plus 1) |
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|{{OEIS link|id=A008864}} |
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|2 |
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|2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, 462, 512, 751, 822, 5312, 7755, 9531, 12379, 15822, 18885, 22971, 23005, 98726, 143018, 151023, 667071, 1195203, 1268979, 1467763, 2013992, 2367906, 3752948, ... |
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|{{OEIS link|id=A002234}} |
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|3 |
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|1, 2, 6, 10, 18, 40, 46, 86, 118, 170, 1172, 1698, 1810, 2268, 4338, 18362, 72662, 88392, 94110, 161538, 168660, 292340, 401208, 560750, 1035092, ... |
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|{{OEIS link|id=A006553}} |
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|4 |
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|1, 2, 3, 5, 8, 14, 23, 63, 107, 132, 428, 530, 1137, 1973, 2000, 7064, 20747, 79574, 113570, 293912, ..., 1993191, ... |
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|{{OEIS link|id=A086661}} |
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|- |
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|5 |
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|8, 14, 42, 384, 564, 4256, 6368, 21132, 27180, 96584, 349656, 545082, ... |
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|{{OEIS link|id=A059676}} |
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|6 |
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|1, 2, 3, 19, 20, 24, 34, 77, 107, 114, 122, 165, 530, 1999, 4359, 11842, 12059, 13802, 22855, 41679, 58185, 145359, 249987, ... |
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|{{OEIS link|id=A059675}} |
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|7 |
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|2, 18, 68, 84, 3812, 14838, 51582, ... |
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|{{OEIS link|id=A242200}} |
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|- |
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|8 |
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|1, 2, 7, 12, 25, 44, 219, 252, 507, 1155, 2259, 2972, 4584, 12422, 13905, 75606, ... |
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|{{OEIS link|id=A242201}} |
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|- |
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|9 |
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|10, 58, 264, 1568, 4198, 24500, ... |
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|{{OEIS link|id=A242202}} |
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|- |
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|10 |
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|2, 3, 8, 11, 15, 39, 60, 72, 77, 117, 183, 252, 396, 1745, 2843, 4665, 5364, ... |
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|{{OEIS link|id=A059671}} |
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|- |
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|11 |
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|2, 8, 252, 1184, 1308, ... |
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|{{OEIS link|id=A299374}} |
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|- |
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|12 |
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|1, 6, 43, 175, 821, 910, 1157, 13748, 27032, 71761, 229918, ... |
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|{{OEIS link|id=A299375}} |
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|- |
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|13 |
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|2, 6, 563528, ... |
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|{{OEIS link|id=A299376}} |
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|- |
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|14 |
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|1, 3, 7, 98, 104, 128, 180, 834, 1633, 8000, 28538, 46605, 131941, 147684, 433734, ... |
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|{{OEIS link|id=A299377}} |
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|- |
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|15 |
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|2, 10, 14, 2312, 16718, 26906, 27512, 41260, 45432, 162454, 217606, ... |
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|{{OEIS link|id=A299378}} |
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|- |
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|16 |
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|167, 189, 639, ... |
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|{{OEIS link|id=A299379}} |
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|- |
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|17 |
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|2, 18, 20, 38, 68, 3122, 3488, 39500, ... |
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|{{OEIS link|id=A299380}} |
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|18 |
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|1, 2, 6, 8, 10, 28, 30, 39, 45, 112, 348, 380, 458, 585, 17559, 38751, 43346, 46984, 92711, ... |
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|{{OEIS link|id=A299381}} |
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|- |
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|19 |
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|12, 410, 33890, 91850, 146478, 189620, 280524, ... |
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|{{OEIS link|id=A299382}} |
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|- |
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|20 |
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|1, 18, 44, 60, 80, 123, 429, 1166, 2065, 8774, 35340, 42968, 50312, 210129, ... |
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|{{OEIS link|id=A299383}} |
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|21 |
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|2, 18, 200, 282, 294, 1174, 2492, 4348, ... |
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| |
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|- |
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|22 |
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|2, 5, 140, 158, 263, 795, 992, 341351, ... |
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| |
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|- |
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|23 |
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|29028, ... |
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| |
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|- |
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|24 |
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|1, 2, 5, 12, 124, 1483, 22075, 29673, 64593, ... |
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| |
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|- |
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|25 |
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|2, 68, 104, 450, ... |
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| |
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|- |
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|26 |
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|3, 8, 79, 132, 243, 373, 720, 1818, 11904, 134778, ... |
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| |
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|27 |
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|10, 18, 20, 2420, 6638, 11368, 14040, 103444, ... |
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| |
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|- |
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|28 |
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|2, 5, 6, 12, 20, 47, 71, 624, 1149, 2399, 8048, 30650, 39161, ... |
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| |
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|- |
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|29 |
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|26850, 237438, 272970, ... |
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| |
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|- |
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|30 |
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|1, 63, 331, 366, 1461, 3493, 4002, 5940, 13572, 34992, 182461, 201038, ... |
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|} |
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{{As of|2018|10}}, the largest known generalized Woodall prime is 17016602×2<sup>17016602</sup> − 1. |
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==See also== |
==See also== |
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==Further reading== |
==Further reading== |
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* {{Citation |first=Richard K. |last=Guy | |
* {{Citation |first=Richard K. |last=Guy |author-link=Richard K. Guy |title=Unsolved Problems in Number Theory |edition=3rd |publisher=[[Springer Verlag]] |location=New York |year=2004 |isbn=0-387-20860-7 |pages=section B20 }}. |
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* {{Citation |first=Wilfrid |last=Keller |title=New Cullen Primes |journal=[[Mathematics of Computation]] |volume=64 |issue=212 |year=1995 |pages=1733–1741 |url=http://www.ams.org/mcom/1995-64-212/S0025-5718-1995-1308456-3/S0025-5718-1995-1308456-3.pdf |doi=10.2307/2153382}}. |
* {{Citation |first=Wilfrid |last=Keller |title=New Cullen Primes |journal=[[Mathematics of Computation]] |volume=64 |issue=212 |year=1995 |pages=1733–1741 |url=http://www.ams.org/mcom/1995-64-212/S0025-5718-1995-1308456-3/S0025-5718-1995-1308456-3.pdf |doi=10.2307/2153382|jstor=2153382 |doi-access=free }}. |
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* {{Citation |first=Chris |last=Caldwell |url=http://primes.utm.edu/top20/page.php?id=7 |title=The Top Twenty: Woodall Primes |work=The [[Prime Pages]] | |
* {{Citation |first=Chris |last=Caldwell |url=http://primes.utm.edu/top20/page.php?id=7 |title=The Top Twenty: Woodall Primes |work=The [[Prime Pages]] |access-date=December 29, 2007 }}. |
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==External links== |
==External links== |
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{{Prime number classes|state=collapsed}} |
{{Prime number classes|state=collapsed}} |
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{{Classes of natural numbers}} |
{{Classes of natural numbers}} |
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__NOTOC__ |
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{{DEFAULTSORT:Woodall Number}} |
{{DEFAULTSORT:Woodall Number}} |
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[[Category:Integer sequences]] |
[[Category:Integer sequences]] |
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[[Category:Unsolved problems in |
[[Category:Unsolved problems in number theory]] |
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[[Category:Classes of prime numbers]] |
Latest revision as of 20:05, 12 December 2024
In number theory, a Woodall number (Wn) is any natural number of the form
for some natural number n. The first few Woodall numbers are:
History
Woodall numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917,[1] inspired by James Cullen's earlier study of the similarly defined Cullen numbers.
Woodall primes
Woodall numbers that are also prime numbers are called Woodall primes; the first few exponents n for which the corresponding Woodall numbers Wn are prime are 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, ... (sequence A002234 in the OEIS); the Woodall primes themselves begin with 7, 23, 383, 32212254719, ... (sequence A050918 in the OEIS).
In 1976 Christopher Hooley showed that almost all Cullen numbers are composite.[2] 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 n · 2n + a + b, where a and b are integers, and in particular, that almost all Woodall numbers are composite.[3] It is an open problem whether there are infinitely many Woodall primes. As of October 2018[update], the largest known Woodall prime is 17016602 × 217016602 − 1.[4] It has 5,122,515 digits and was found by Diego Bertolotti in March 2018 in the distributed computing project PrimeGrid.[5]
Restrictions
Starting with W4 = 63 and W5 = 159, every sixth Woodall number is divisible by 3; thus, in order for Wn to be prime, the index n cannot be congruent to 4 or 5 (modulo 6). Also, for a positive integer m, the Woodall number W2m may be prime only if 2m + 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 is +1 and
- W(3p − 1) / 2 if the Jacobi symbol is −1.[citation needed]
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[6]
- 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, ... (sequence A240235 in the OEIS)
As of November 2021[update], the largest known generalized Woodall prime with base greater than 2 is 2740879 × 322740879 − 1.[7]
See also
- Mersenne prime - Prime numbers of the form 2n − 1.
References
- ^ Cunningham, A. J. C; Woodall, H. J. (1917), "Factorisation of and ", Messenger of Mathematics, 47: 1–38.
- ^ Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs. Vol. 104. Providence, RI: American Mathematical Society. p. 94. ISBN 0-8218-3387-1. Zbl 1033.11006.
- ^ Keller, Wilfrid (January 1995). "New Cullen primes". Mathematics of Computation. 64 (212): 1739. doi:10.1090/S0025-5718-1995-1308456-3. ISSN 0025-5718. Keller, Wilfrid (December 2013). "Wilfrid Keller". www.fermatsearch.org. Hamburg. Archived from the original on February 28, 2020. Retrieved October 1, 2020.
- ^ "The Prime Database: 8508301*2^17016603-1", Chris Caldwell's The Largest Known Primes Database, retrieved March 24, 2018
- ^ PrimeGrid, Announcement of 17016602*2^17016602 - 1 (PDF), retrieved April 1, 2018
- ^ List of generalized Woodall primes base 3 to 10000
- ^ "The Top Twenty: Generalized Woodall". primes.utm.edu. Retrieved 20 November 2021.
Further reading
- Guy, Richard K. (2004), Unsolved Problems in Number Theory (3rd ed.), New York: Springer Verlag, pp. section B20, ISBN 0-387-20860-7.
- Keller, Wilfrid (1995), "New Cullen Primes" (PDF), Mathematics of Computation, 64 (212): 1733–1741, doi:10.2307/2153382, JSTOR 2153382.
- Caldwell, Chris, "The Top Twenty: Woodall Primes", The Prime Pages, retrieved December 29, 2007.
External links
- Chris Caldwell, The Prime Glossary: Woodall number, and The Top Twenty: Woodall, and The Top Twenty: Generalized Woodall, at The Prime Pages.
- Weisstein, Eric W. "Woodall number". MathWorld.
- Steven Harvey, List of Generalized Woodall primes.
- Paul Leyland, Generalized Cullen and Woodall Numbers