Difference between revisions of "CarolKynea prime"
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==History==  ==History==  
−  ==  +  ==Data== 
+  ===All bases===  
+  All bases with their own page are listed here:  
+  {{#dpl:  
+  debug=1  
+  category=CarolKynea_prime  
+  titlematch=CarolKynea prime %  
+  mode=userformat  
+  resultsheader=There are <b>%PAGES%</b> sequences.\n  
+  }}  
*[[CarolKynea table]]  *[[CarolKynea table]]  
+  
+  ===Bases which are a power of===  
+  {{#dpl:  
+  debug=1  
+  category=CarolKynea powerof  
+  titlematch=CarolKynea prime %  
+  mode=userformat  
+  resultsheader=There are <b>%PAGES%</b> sequences.\n  
+  noresultsheader=There are no entries.\n  
+  }}  
+  *[[:Category:CarolKynea powerofCategory]]  
+  
+  ===Bases without a Carol prime===  
+  {{#dpl:  
+  debug=1  
+  category=CarolKynea without Carol  
+  titlematch=CarolKynea prime %  
+  mode=userformat  
+  resultsheader=There are <b>%PAGES%</b> sequences.\n  
+  noresultsheader=There are no entries.\n  
+  }}  
+  *[[:Category:CarolKynea without CarolCategory]]  
+  
+  ===Bases without a Kynea prime===  
+  {{#dpl:  
+  debug=1  
+  category=CarolKynea without Kynea  
+  titlematch=CarolKynea prime %  
+  mode=userformat  
+  resultsheader=There are <b>%PAGES%</b> sequences.\n  
+  noresultsheader=There are no entries.\n  
+  }}  
+  *[[:Category:CarolKynea without KyneaCategory]]  
+  
+  ===Bases without a Carol and Kynea prime===  
+  {{#dpl:  
+  debug=1  
+  category=CarolKynea without both  
+  titlematch=CarolKynea prime %  
+  mode=userformat  
+  resultsheader=There are <b>%PAGES%</b> sequences.\n  
+  noresultsheader=There are no entries.\n  
+  }}  
+  *[[:Category:CarolKynea without bothCategory]]  
+  
+  ===Remaining data===  
+  All data not yet given by an own page can be found <b>here</b>.  
==External links==  ==External links== 
Revision as of 11:10, 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 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 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.
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 
