Difference between revisions of "CarolKynea prime"
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*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.  *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<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.  
−  *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 not necessary to search these bases separately.  +  *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<sup>2</sup>k).  Due to the form of these numbers, they are also classified as nearsquare numbers (numbers of the form n<sup>2</sup>k).  
Revision as of 12:54, 11 June 2019
Contents
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
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
Number classes
General numbers 
Special numbers 
Prime numbers 
