CarolKynea prime
目錄
Definitions
In the context of the Carol/Kynea prime search, a Carol number is a number of the form [math](b^n1)^22[/math] and a Kynea number is a number of the form [math](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](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]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, who named them after personal acquaintances. 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. 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 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 
(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 
(2^{621443}+1)^{2}2  374146  Mark Rodenkirch  20160530 
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 74 sequences.
Bases which are a power of
There are 13 sequences.
Bases without a Carol prime
There are 2 sequences.
Bases without a Kynea prime
There are 2 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.
External links
 NearSquare primes
 Carol number
 Kynea number
 Reservation thread
 Primes and results
 More data
 Old project by S.Harvey
General numbers 
Special numbers 
Prime numbers 
