Difference between revisions of "Carol-Kynea prime"

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*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.
 
*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 (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.
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*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 n<sup>2</sup>-k).
 
Due to the form of these numbers, they are also classified as near-square numbers (numbers of the form n<sup>2</sup>-k).
  

Revision as of 12:54, 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 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 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

Number classes
General numbers
Special numbers
Prime numbers