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Difference between revisions of "Cullen number"
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A '''Cullen number ''C<sub>n</sub>'' ''' is a number of the form ''n'' × 2<sup>''n''</sup> + 1,<br /> | A '''Cullen number ''C<sub>n</sub>'' ''' is a number of the form ''n'' × 2<sup>''n''</sup> + 1,<br /> | ||
a '''generalized Cullen number''' base ''b'' is a number of the form ''n'' × ''b''<sup>''n''</sup> + 1. '''(perhaps own page?)''' | a '''generalized Cullen number''' base ''b'' is a number of the form ''n'' × ''b''<sup>''n''</sup> + 1. '''(perhaps own page?)''' | ||
==Cullen primes== | ==Cullen primes== | ||
| − | ''C<sub>n</sub>'' is [[prime]] for ''n'' = 1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, {{PP|87775|6328548}}, {{PP|89536|6679881}} and for no other ''n'' < {{Num|17200000}} ({{OEIS|l|A005849}}). | + | ''C<sub>n</sub>'' is [[prime]] for ''n'' = 1, 141, {{PP|40087|4713}}, {{PP|38112|5795}}, {{PP|37374|6611}}, {{PP|23436|18496}}, {{PP|17877|32292}}, {{PP|17850|32469}}, {{PP|10752|59656}}, {{PP|5931|90825}}, {{PP|271|262419}}, {{PP|102|361275}}, {{PP|77|481899}}, {{PP|75433|1354828}}, {{PP|87775|6328548}}, {{PP|89536|6679881}} and for no other ''n'' < {{Num|17200000}} ({{OEIS|l|A005849}}). |
The largest known Cullen prime is {{PP|89536|{{Num|6679881}} × 2<sup>{{Num|6679881}}</sup> + 1}} and the largest known generalized Cullen prime is {{PP|124515|{{Num|1806676}} × 41<sup>{{Num|1806676}}</sup> + 1}}. | The largest known Cullen prime is {{PP|89536|{{Num|6679881}} × 2<sup>{{Num|6679881}}</sup> + 1}} and the largest known generalized Cullen prime is {{PP|124515|{{Num|1806676}} × 41<sup>{{Num|1806676}}</sup> + 1}}. | ||
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*[http://web.archive.org/web/20070626212719/http://www.primzahlenarchiv.de/index.html Web archive "primzahlenarchiv.de", Daniel Hermle (101 ≤ base ≤ 200), dated 2007] | *[http://web.archive.org/web/20070626212719/http://www.primzahlenarchiv.de/index.html Web archive "primzahlenarchiv.de", Daniel Hermle (101 ≤ base ≤ 200), dated 2007] | ||
*[http://harvey563.tripod.com/GClist.txt Generalized Cullen, Steven Harvey (101 ≤ base ≤ 10000), dated 2017] | *[http://harvey563.tripod.com/GClist.txt Generalized Cullen, Steven Harvey (101 ≤ base ≤ 10000), dated 2017] | ||
| − | *[[Wikipedia:Cullen_number| | + | *[[Wikipedia:Cullen_number|Cullen number]] |
*[http://mathworld.wolfram.com/CullenNumber.html Mathworld] | *[http://mathworld.wolfram.com/CullenNumber.html Mathworld] | ||
*[http://primes.utm.edu/glossary/xpage/Cullens.html The Prime Glossary] | *[http://primes.utm.edu/glossary/xpage/Cullens.html The Prime Glossary] | ||
[[Category:Numbers]] | [[Category:Numbers]] | ||
Revision as of 10:09, 6 March 2019
| Last update: 2019-02-27 |
A Cullen number Cn is a number of the form n × 2n + 1,
a generalized Cullen number base b is a number of the form n × bn + 1. (perhaps own page?)
Cullen primes
Cn is prime for n = 1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881 and for no other n < 17,200,000 (sequence A005849 in OEIS).
The largest known Cullen prime is 6,679,881 × 26,679,881 + 1 and the largest known generalized Cullen prime is 1,806,676 × 411,806,676 + 1.
Current status
See also
External links
- Top 20 Cullen Primes
- Top 20 Generalized Cullen Primes
- The Chronology of Prime Number Records
- Web archive "prothsearch.net"
- Generalized Cullen Search, Günter Löh (3 ≤ base ≤ 100), dated Sept. 2017
- Web archive "primzahlenarchiv.de", Daniel Hermle (101 ≤ base ≤ 200), dated 2007
- Generalized Cullen, Steven Harvey (101 ≤ base ≤ 10000), dated 2017
- Cullen number
- Mathworld
- The Prime Glossary
