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CarolKynea prime
Contents
Definitions
In the context of the Carol/Kynea prime search, a Carol number is a number of the form [math]\displaystyle{ (b^n1)^22 }[/math] and a Kynea number is a number of the form [math]\displaystyle{ (b^n+1)^22 }[/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)^22 }[/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^{n}±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 (b^{n}±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 nearsquare numbers (numbers of the form n^{2}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 20151226 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 

(290^{124116}1)^{2}2  611246  Karsten Bonath  20190301 
(2^{695631}1)^{2}2  418812  Mark Rodenkirch  20160716 
(2^{688042}1)^{2}2  414243  Mark Rodenkirch  20160705 
(178^{87525}1)^{2}2  393937  Serge Batalov  20160521 
(2^{653490}1)^{2}2  393441  Mark Rodenkirch  20160603 
Top 5 Kynea primes
Prime  Digits  Found by  Date 

(362^{133647}+1)^{2}2  683928  Karsten Bonath  20190617 
(2^{852770}+1)^{2}2  513419  Ryan Propper  20190714 
(30^{157950}+1)^{2}2  466623  Serge Batalov  20160522 
(2^{661478}+1)^{2}2  398250  Mark Rodenkirch  20160618 
(1968^{58533}+1)^{2}2  385619  Clint Stillman  20171130 
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 62 sequences.
Bases without a Kynea prime
There are 62 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, nrange=110000, max prime factor 10^{9}). 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 10^{12}, 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"(12^688351)^22"
.
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
 NearSquare primes
 Carol number
 Kynea number
 Old thread
 More data for bases ≤ 3000 and n ≤ 2000
 Old project by S.Harvey
General numbers 
Special numbers 
Prime numbers 
