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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. | + | *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
Contents
Definitions
In the context of the Carol/Kynea prime search, a Carol number is a number of the form [math]\displaystyle{ (b^n-1)^2-2 }[/math] and a Kynea number is a number of the form [math]\displaystyle{ (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]\displaystyle{ (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]\displaystyle{ 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 382 sequences.
Bases which are a power of
There are 22 sequences.
Bases without a Carol prime
There are 62 sequences.
Bases without a Kynea prime
There are 61 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
- Near-Square primes
- Carol number
- Kynea number
- Reservation thread
- Primes and results
- More data
- Old project by S.Harvey
Number classes
| General numbers |
| Special numbers |
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| Prime numbers |
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