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Difference between revisions of "Carol-Kynea prime"
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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== |
Revision as of 15:04, 4 July 2019
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] (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]\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 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 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 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). 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
- Search maintained by Gary Barnes
- Reservation thread
- Primes and results thread
Others
- Near-Square primes
- Carol number
- Kynea number
- Old thread
- More data for bases ≤ 3000 and n ≤ 2000
- Old project by S.Harvey
General numbers |
Special numbers |
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Prime numbers |
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