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Difference between revisions of "Carol-Kynea prime"
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==Definitions== | ==Definitions== | ||
| − | In the context of the Carol/Kynea prime search, a Carol number is a number of the form <math>(b^n-1)^2-2</math> and a Kynea number is a number of the form <math>(b^n+1)^2-2</math> | + | In the context of the Carol/Kynea prime search, a Carol number is a number of the form <math>(b^n-1)^2-2</math> and a Kynea number is a number of the form <math>(b^n+1)^2-2</math>. A Carol/Kynea prime is a [[prime]] which has one of the above forms. A prime of these forms must satisfy the following criteria: |
*b must be even, since if it is odd then <math>(b^n±1)^2-2</math> is always even, and thus can’t be prime. | *b must be even, since if it is odd then <math>(b^n±1)^2-2</math> is always even, and thus can’t be prime. | ||
*n must be greater than or equal to 1. For any b, if n is 0 then (b<sup>n</sup>±1)<sup>2</sup> is equal to 1, and thus yields -1 when 2 is subtracted from it. By definition -1 is not prime. If n is negative then (b<sup>n</sup>±1)<sup>2</sup> is not necessarily an integer. | *n must be greater than or equal to 1. For any b, if n is 0 then (b<sup>n</sup>±1)<sup>2</sup> is equal to 1, and thus yields -1 when 2 is subtracted from it. By definition -1 is not prime. If n is negative then (b<sup>n</sup>±1)<sup>2</sup> is not necessarily an integer. | ||
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Carol and Kynea numbers were first studied by [[Cletus Emmanuel]] in 1995<ref>[https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/5099 Yahoo Group "Primenumbers", 2002-02-04]</ref>, who named them after personal acquaintances<ref>[https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/14584 Yahoo Group "Primenumbers", 2004-02-23]</ref>. He searched these forms for primes up to the limit of 15000. | Carol and Kynea numbers were first studied by [[Cletus Emmanuel]] in 1995<ref>[https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/5099 Yahoo Group "Primenumbers", 2002-02-04]</ref>, who named them after personal acquaintances<ref>[https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/14584 Yahoo Group "Primenumbers", 2004-02-23]</ref>. He searched these forms for primes up to the limit of 15000. | ||
| − | Starting in 2004, [[Steven Harvey]] maintained a search for this form. At this time [[Multisieve]] and [[cksieve]] were used to sieve these forms and [[PFGW]] was used to test for primality. The search went dormant in 2011 and was resurrected in 2015 by [[Mark Rodenkirch]]. Initially Multisieve was used, but then later on he wrote cksieve which would later become part of [[Mtsieve]] framework. | + | Starting in 2004, [[Steven Harvey]] maintained a search for this form. At this time [[Multisieve]] and [[cksieve]] were used to sieve these forms and [[PFGW]] was used to test for primality. The search went dormant in 2011 and was resurrected in 2015 by [[Mark Rodenkirch]]. Initially Multisieve was used, but then later on he wrote cksieve which would later become part of the [[Mtsieve]] framework. |
On 2015-12-26 Mark opened a thread<ref>[https://www.mersenneforum.org/showthread.php?t=20784 Thread "Carol / Kynea search (Near-power primes)"]</ref> for a coordinated search of Carol/Kynea numbers on [[MersenneForum]], which continues to this day (although now [[Gary Barnes]], maintainer of [[No Prime Left Behind|NPLB]] and [[Conjectures 'R Us|CRUS]], maintains the search). | On 2015-12-26 Mark opened a thread<ref>[https://www.mersenneforum.org/showthread.php?t=20784 Thread "Carol / Kynea search (Near-power primes)"]</ref> for a coordinated search of Carol/Kynea numbers on [[MersenneForum]], which continues to this day (although now [[Gary Barnes]], maintainer of [[No Prime Left Behind|NPLB]] and [[Conjectures 'R Us|CRUS]], maintains the search). | ||
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|- | |- | ||
|{{T5000|126269|(290<sup>124116</sup>-1)<sup>2</sup>-2}}||611246||[[Karsten Bonath]]||2019-03-01 | |{{T5000|126269|(290<sup>124116</sup>-1)<sup>2</sup>-2}}||611246||[[Karsten Bonath]]||2019-03-01 | ||
| + | |- | ||
| + | |{{T5000|131572|(968<sup>79512</sup>-1)<sup>2</sup>-2}}||474826||[[Karsten Bonath]]||2021-01-13 | ||
|- | |- | ||
|{{T5000|121905|(2<sup>695631</sup>-1)<sup>2</sup>-2}}||418812||[[Mark Rodenkirch]]||2016-07-16 | |{{T5000|121905|(2<sup>695631</sup>-1)<sup>2</sup>-2}}||418812||[[Mark Rodenkirch]]||2016-07-16 | ||
| Line 24: | Line 26: | ||
|- | |- | ||
|{{T5000|121680|(178<sup>87525</sup>-1)<sup>2</sup>-2}}||393937||[[Serge Batalov]]||2016-05-21 | |{{T5000|121680|(178<sup>87525</sup>-1)<sup>2</sup>-2}}||393937||[[Serge Batalov]]||2016-05-21 | ||
| − | |||
| − | |||
|} | |} | ||
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|- | |- | ||
|{{T5000|126558|(362<sup>133647</sup>+1)<sup>2</sup>-2}}||683928||[[Karsten Bonath]]||2019-06-17 | |{{T5000|126558|(362<sup>133647</sup>+1)<sup>2</sup>-2}}||683928||[[Karsten Bonath]]||2019-06-17 | ||
| + | |- | ||
| + | |{{T5000|126646|(2<sup>852770</sup>+1)<sup>2</sup>-2}}||513419||[[Ryan Propper]]||2019-07-14 | ||
|- | |- | ||
|{{T5000|121686|(30<sup>157950</sup>+1)<sup>2</sup>-2}}||466623||[[Serge Batalov]]||2016-05-22 | |{{T5000|121686|(30<sup>157950</sup>+1)<sup>2</sup>-2}}||466623||[[Serge Batalov]]||2016-05-22 | ||
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|- | |- | ||
|(1968<sup>58533</sup>+1)<sup>2</sup>-2||385619||[[Clint Stillman]]||2017-11-30 | |(1968<sup>58533</sup>+1)<sup>2</sup>-2||385619||[[Clint Stillman]]||2017-11-30 | ||
| − | |||
| − | |||
|} | |} | ||
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===Remaining data=== | ===Remaining data=== | ||
| − | All | + | All bases <= 3000 not listed yet on their own page are tested to n=2000 and listed in a file. |
| + | |||
| + | The data file can be found [[:File:Carol-Kynea prime remaining.csv|here]]. | ||
| + | {{#get_web_data:url=https://www.rieselprime.de/z/images/5/50/Carol-Kynea_prime_remaining.csv|format=csv with header|delimiter=;|data=b=b,C=C,K=K}} | ||
| + | <div style="width:46em; height:500px; overflow:auto;"> | ||
| + | {| class="wikitable sortable" style="height: 200px" | ||
| + | ! data-sort-type="number" class="fixhead" | Base | ||
| + | ! class="fixhead unsortable"| Carol primes | ||
| + | ! class="fixhead unsortable"| Kynea primes | ||
| + | {{#for_external_table:<nowiki/> | ||
| + | {{!}}- | ||
| + | {{!}} style="text-align:right" {{!}} {{{b}}} | ||
| + | {{!}} style="text-align:left" {{!}} {{{C}}} | ||
| + | {{!}} style="text-align:left" {{!}} {{{K}}} | ||
| + | }} | ||
| + | |}</div> | ||
==How to participate?== | ==How to participate?== | ||
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===Sieving=== | ===Sieving=== | ||
*Use [[cksieve]] (from [[Mtsieve]]) and | *Use [[cksieve]] (from [[Mtsieve]]) and | ||
| − | **run a new sieve by calling <code>cksieve -b 12 -n 1 -N 10000 -P 1000000000</code> (for base=12, n-range=1-10000, max prime factor 10<sup>9</sup>). | + | **run a new sieve by calling <code>cksieve -b 12 -n 1 -N 10000 -P 1000000000</code> (for base=12, n-range=1-10000, max prime factor 10<sup>9</sup>). The sieve file will be written to ck_12.pfgw. |
**rerun an old sieve by calling <code>cksieve -P 1e12 -i ck_12.pfgw -o ck_12.pfgw -f factors.txt</code> (for base=12, max prime factor 10<sup>12</sup>, input/output files given, storing factors to "factors.txt"). | **rerun an old sieve by calling <code>cksieve -P 1e12 -i ck_12.pfgw -o ck_12.pfgw -f factors.txt</code> (for base=12, max prime factor 10<sup>12</sup>, input/output files given, storing factors to "factors.txt"). | ||
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===Prime testing=== | ===Prime testing=== | ||
After testing with PFGW higher probable primes will be written in "pfgw.log". These have to be checked prime by calling like <code>pfgw64 -tp -q"(12^68835-1)^2-2"</code>. | After testing with PFGW higher probable primes will be written in "pfgw.log". These have to be checked prime by calling like <code>pfgw64 -tp -q"(12^68835-1)^2-2"</code>. | ||
| + | |||
| + | ===Reporting=== | ||
| + | Once you have completed your range, report any primes found in this [https://www.mersenneforum.org/showthread.php?t=21251 thread]. Then report the completed range in the reservation thread and specify whether you will continue with the base or release it. | ||
==References== | ==References== | ||
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==External links== | ==External links== | ||
===Current=== | ===Current=== | ||
| − | *[http://www.noprimeleftbehind.net/Carol-Kynea-prime-search.htm | + | *[http://www.noprimeleftbehind.net/Carol-Kynea-prime-search.htm Results] maintained by [[Gary Barnes]] |
| − | *[https://www.mersenneforum.org/showthread.php?t=21216 | + | *[https://www.mersenneforum.org/showthread.php?t=21216 "Carol / Kynea Coordinated Search - Reservations"] at [[MersenneForum]] up to post #247 (2020-05-31) |
| − | *[https://www.mersenneforum.org/showthread.php?t=21251 Primes | + | *[https://www.mersenneforum.org/showthread.php?t=21251 "Carol / Kynea Primes"] at [[MersenneForum]] up to post #239 (2020-04-08 ) |
===Others=== | ===Others=== | ||
*[http://mathworld.wolfram.com/Near-SquarePrime.html Near-Square primes] | *[http://mathworld.wolfram.com/Near-SquarePrime.html Near-Square primes] | ||
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*[[Wikipedia:Kynea_number|Kynea number]] | *[[Wikipedia:Kynea_number|Kynea number]] | ||
*[https://www.mersenneforum.org/showthread.php?t=20784 Old thread] | *[https://www.mersenneforum.org/showthread.php?t=20784 Old thread] | ||
| − | |||
*[http://harvey563.tripod.com/Carol_Kynea.txt Old project by S.Harvey] | *[http://harvey563.tripod.com/Carol_Kynea.txt Old project by S.Harvey] | ||
{{Navbox NumberClasses}} | {{Navbox NumberClasses}} | ||
[[Category:Carol-Kynea prime| ]] | [[Category:Carol-Kynea prime| ]] | ||
Revision as of 08:26, 14 January 2021
Contents
Definitions
In the context of the Carol/Kynea prime search, a Carol number is a number of the form [math]\displaystyle{ (b^n-1)^2-2 }[/math] and a Kynea number is a number of the form [math]\displaystyle{ (b^n+1)^2-2 }[/math]. A Carol/Kynea prime is a prime which has one of the above forms. A prime of these forms must satisfy the following criteria:
- b must be even, since if it is odd then [math]\displaystyle{ (b^n±1)^2-2 }[/math] is always even, and thus can’t be prime.
- n must be greater than or equal to 1. For any b, if n is 0 then (bn±1)2 is equal to 1, and thus yields -1 when 2 is subtracted from it. By definition -1 is not prime. If n is negative then (bn±1)2 is not necessarily an integer.
- b may be a perfect power of another integer. However these form a subset of another base’s primes (ex. Base 4 Carol/Kynea primes are Base 2 Carol/Kynea primes where [math]\displaystyle{ n \bmod 2 \equiv 0 }[/math]). So it is not necessary to search these bases separately.
Due to the form of these numbers, they are also classified as near-square numbers (numbers of the form n2-k).
History
Carol and Kynea numbers were first studied by Cletus Emmanuel in 1995[1], who named them after personal acquaintances[2]. He searched these forms for primes up to the limit of 15000.
Starting in 2004, Steven Harvey maintained a search for this form. At this time Multisieve and cksieve were used to sieve these forms and PFGW was used to test for primality. The search went dormant in 2011 and was resurrected in 2015 by Mark Rodenkirch. Initially Multisieve was used, but then later on he wrote cksieve which would later become part of the Mtsieve framework.
On 2015-12-26 Mark opened a thread[3] for a coordinated search of Carol/Kynea numbers on MersenneForum, which continues to this day (although now Gary Barnes, maintainer of NPLB and CRUS, maintains the search).
Top 5 Carol primes
| Prime | Digits | Found by | Date |
|---|---|---|---|
| (290124116-1)2-2 | 611246 | Karsten Bonath | 2019-03-01 |
| (96879512-1)2-2 | 474826 | Karsten Bonath | 2021-01-13 |
| (2695631-1)2-2 | 418812 | Mark Rodenkirch | 2016-07-16 |
| (2688042-1)2-2 | 414243 | Mark Rodenkirch | 2016-07-05 |
| (17887525-1)2-2 | 393937 | Serge Batalov | 2016-05-21 |
Top 5 Kynea primes
| Prime | Digits | Found by | Date |
|---|---|---|---|
| (362133647+1)2-2 | 683928 | Karsten Bonath | 2019-06-17 |
| (2852770+1)2-2 | 513419 | Ryan Propper | 2019-07-14 |
| (30157950+1)2-2 | 466623 | Serge Batalov | 2016-05-22 |
| (2661478+1)2-2 | 398250 | Mark Rodenkirch | 2016-06-18 |
| (196858533+1)2-2 | 385619 | Clint Stillman | 2017-11-30 |
OEIS sequences
These are available OEIS sequences:
| Base | Carol | Kynea |
|---|---|---|
| 2 | A091515 | A091513 |
| 6 | A100901 | A100902 |
| 10 | A100903 | A100904 |
| 14 | A100905 | A100906 |
| 22 | A100907 | A100908 |
Data
All bases
All bases with their own page are listed here: There are 382 sequences.
Bases which are a power of
There are 22 sequences.
Bases without a Carol prime
There are 62 sequences.
Bases without a Kynea prime
There are 61 sequences.
Bases without a Carol and Kynea prime
There are 1 sequences.
Remaining data
All bases <= 3000 not listed yet on their own page are tested to n=2000 and listed in a file.
The data file can be found here.
Error while fetching data from URL https://www.rieselprime.de/z/images/5/50/Carol-Kynea_prime_remaining.csv: $2.
Error fetching URL: error:0A000410:SSL routines::sslv3 alert handshake failure
Could not get URL https://www.rieselprime.de/z/images/5/50/Carol-Kynea_prime_remaining.csv after 3 tries.
| Base | Carol primes | Kynea primes |
|---|
How to participate?
Reserving
- Reserve your base(s)/range(s) in this thread.
Sieving
- Use cksieve (from Mtsieve) and
- run a new sieve by calling
cksieve -b 12 -n 1 -N 10000 -P 1000000000(for base=12, n-range=1-10000, max prime factor 109). The sieve file will be written to ck_12.pfgw. - rerun an old sieve by calling
cksieve -P 1e12 -i ck_12.pfgw -o ck_12.pfgw -f factors.txt(for base=12, max prime factor 1012, input/output files given, storing factors to "factors.txt").
- run a new sieve by calling
PRP testing
- Use PFGW calling
pfgw64.exe -f0 ck_12.pfgw(running candidates file for base 12, no further factoring).
Prime testing
After testing with PFGW higher probable primes will be written in "pfgw.log". These have to be checked prime by calling like pfgw64 -tp -q"(12^68835-1)^2-2".
Reporting
Once you have completed your range, report any primes found in this thread. Then report the completed range in the reservation thread and specify whether you will continue with the base or release it.
References
External links
Current
- Results maintained by Gary Barnes
- "Carol / Kynea Coordinated Search - Reservations" at MersenneForum up to post #247 (2020-05-31)
- "Carol / Kynea Primes" at MersenneForum up to post #239 (2020-04-08 )
Others
Number classes
| General numbers |
| Special numbers |
|
| Prime numbers |
|
