Difference between revisions of "Carol-Kynea prime"

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===Bases which are a power of===
 
===Bases which are a power of===
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There are <b>{{PAGESINCATEGORY:Carol-Kynea power-of}}</b> sequences.
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|category=Carol-Kynea power-of
 
|titlematch=Carol-Kynea prime %
 
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|resultsheader=There are <b>%PAGES%</b> sequences.\n
 
|noresultsheader=There are no entries.\n
 
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*[[:Category:Carol-Kynea power-of|Category]]
 
*[[:Category:Carol-Kynea power-of|Category]]
  
 
===Bases without a Carol prime===
 
===Bases without a Carol prime===
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There are <b>{{PAGESINCATEGORY:Carol-Kynea without Carol}}</b> sequences.
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|category=Carol-Kynea without Carol
 
|titlematch=Carol-Kynea prime %
 
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|resultsheader=There are <b>%PAGES%</b> sequences.\n
 
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*[[:Category:Carol-Kynea without Carol|Category]]
 
*[[:Category:Carol-Kynea without Carol|Category]]
  
 
===Bases without a Kynea prime===
 
===Bases without a Kynea prime===
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There are <b>{{PAGESINCATEGORY:Carol-Kynea without Kynea}}</b> sequences.
|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
 
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*[[:Category:Carol-Kynea without Kynea|Category]]
 
*[[:Category:Carol-Kynea without Kynea|Category]]
  
 
===Bases without a Carol and Kynea prime===
 
===Bases without a Carol and Kynea prime===
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There are <b>{{PAGESINCATEGORY:Carol-Kynea without both}}</b> sequences.
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|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]]
 
*[[:Category:Carol-Kynea without both|Category]]
  

Revision as of 14:13, 13 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 is 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