Difference between revisions of "Carol-Kynea prime"

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==History==
 
==History==
  
==See also==
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==Data==
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===All bases===
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All bases with their own page are listed here:
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{{#dpl:
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|debug=1
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|category=Carol-Kynea_prime
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|titlematch=Carol-Kynea prime %
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|mode=userformat
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|resultsheader=There are <b>%PAGES%</b> sequences.\n
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}}
 
*[[Carol-Kynea table]]
 
*[[Carol-Kynea table]]
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 +
===Bases which are a power of===
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{{#dpl:
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|debug=1
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|category=Carol-Kynea power-of
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|titlematch=Carol-Kynea prime %
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|mode=userformat
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|resultsheader=There are <b>%PAGES%</b> sequences.\n
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|noresultsheader=There are no entries.\n
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}}
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*[[:Category:Carol-Kynea power-of|Category]]
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 +
===Bases without a Carol prime===
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{{#dpl:
 +
|debug=1
 +
|category=Carol-Kynea without Carol
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|titlematch=Carol-Kynea prime %
 +
|mode=userformat
 +
|resultsheader=There are <b>%PAGES%</b> sequences.\n
 +
|noresultsheader=There are no entries.\n
 +
}}
 +
*[[:Category:Carol-Kynea without Carol|Category]]
 +
 +
===Bases without a Kynea prime===
 +
{{#dpl:
 +
|debug=1
 +
|category=Carol-Kynea without Kynea
 +
|titlematch=Carol-Kynea prime %
 +
|mode=userformat
 +
|resultsheader=There are <b>%PAGES%</b> sequences.\n
 +
|noresultsheader=There are no entries.\n
 +
}}
 +
*[[:Category:Carol-Kynea without Kynea|Category]]
 +
 +
===Bases without a Carol and Kynea prime===
 +
{{#dpl:
 +
|debug=1
 +
|category=Carol-Kynea without both
 +
|titlematch=Carol-Kynea prime %
 +
|mode=userformat
 +
|resultsheader=There are <b>%PAGES%</b> sequences.\n
 +
|noresultsheader=There are no entries.\n
 +
}}
 +
*[[:Category:Carol-Kynea without both|Category]]
 +
 +
===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

Definitions

In the context of the Carol/Kynea prime search, a Carol number is a number of the form [math](b^n-1)^2-2[/math] and a Kynea number is a number of the form [math](b^n+1)^2-2[/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)^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]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 near-square numbers (numbers of the form n2-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

Number classes
General numbers
Special numbers
Prime numbers