Difference between revisions of "Carol-Kynea prime"

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(draft of history (to do: history between start of Emmanuel’s search and Harvey’s efforts?))
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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>. A Carol/Kynea prime is a [[prime]] which has one of the above forms. A prime of these forms must satisfy the following criteria:
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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> (can be written also as 4<sup>n</sup> ± 2<sup>n+1</sup>-1). 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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==History==
 
==History==
Carol and Kynea numbers were first studied by [[Cletus Emmanuel]], who named them after personal acquaintances. He searched these forms for primes up to the limit of 15000.
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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.
Starting in 2004, S. Harvey maintained a search for this form. At this time [[Multisieve]] and ck 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.
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On April 15, 2016 Mark opened a thread 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 %27R Us|CRUS]], maintains the search).
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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.
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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).
  
 
==Top 5 Carol primes==
 
==Top 5 Carol primes==
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===Remaining data===
 
===Remaining data===
 
All data not yet given by an own page can be found <b>here</b>.
 
All data not yet given by an own page can be found <b>here</b>.
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 +
==How to participate?==
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===Reserving===
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*Reserve your base(s)/range(s) in this [https://www.mersenneforum.org/showthread.php?t=21216 thread].
 +
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===Sieving===
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*Use [[cksieve]] (from [[Mtsieve]]) and
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**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>).
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**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").
 +
 +
===PRP testing===
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*Use [[PFGW]] calling <code>pfgw64.exe -l 0 ck_12.pfgw</code> (running candidates file for base 12, no further factoring).
 +
 +
===Prime testing===
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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>.
 +
 +
==References==
 +
<references/>
  
 
==External links==
 
==External links==

Revision as of 08:17, 1 July 2019

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] (can be written also as 4n ± 2n+1-1). 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.
  • 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]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 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
(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
(2653490-1)2-2 393441 Mark Rodenkirch 2016-06-03

Top 5 Kynea primes

Prime Digits Found by Date
(362133647+1)2-2 683928 Karsten Bonath 2019-06-17
(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
(2621443+1)2-2 374146 Mark Rodenkirch 2016-05-30

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 381 sequences.

Bases which are a power of

There are 22 sequences.

Bases without a Carol prime

There are 90 sequences.

Bases without a Kynea prime

There are 75 sequences.

Bases without a Carol and Kynea prime

There are 2 sequences.

Remaining data

All data not yet given by an own page can be found here.

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).
    • 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").

PRP testing

  • Use PFGW calling pfgw64.exe -l 0 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".

References

External links

Number classes
General numbers
Special numbers
Prime numbers