Currently there may be errors shown on top of a page, because of a missing Wiki update (PHP version and extension DPL3).
Navigation
Topics Help • Register • News • History • How to • Sequences statistics • Template prototypes

Difference between revisions of "Carol-Kynea prime"

From Prime-Wiki
Jump to: navigation, search
(countings)
(Top5 added)
Line 7: Line 7:
  
 
==History==
 
==History==
 +
 +
==Top 5 Carol primes==
 +
{| class="wikitable"
 +
! Prime !! Digits !! Found by !! Date
 +
|-
 +
|{{T5000|126269|(290<sup>124116</sup>-1)<sup>2</sup>-2}}||611246||[[Karsten Bonath]]||2019-03-01
 +
|-
 +
|{{T5000|121905|(2<sup>695631</sup>-1)<sup>2</sup>-2}}||418812||[[Mark Rodenkirch]]||2016-07-16
 +
|-
 +
|{{T5000|121867|(2<sup>688042</sup>-1)<sup>2</sup>-2}}||414243||[[Mark Rodenkirch]]||2016-07-05
 +
|-
 +
|{{T5000|121680|(178<sup>87525</sup>-1)<sup>2</sup>-2}}||393937||[[Serge Batalov]]||2016-05-21
 +
|-
 +
|{{T5000|121779|(2<sup>653490</sup>-1)<sup>2</sup>-2}}||393441||[[Mark Rodenkirch]]||2016-06-03
 +
|}
 +
 +
==Top 5 Kynea primes==
 +
{| class="wikitable"
 +
! Prime !! Digits !! Found by !! Date
 +
|-
 +
|{{T5000|126558|(362<sup>133647</sup>+1)<sup>2</sup>-2}}||683928||[[Karsten Bonath]]||2019-06-17
 +
|-
 +
|{{T5000|121686|(30<sup>157950</sup>+1)<sup>2</sup>-2}}||466623||[[Serge Batalov]]||2016-05-22
 +
|-
 +
|{{T5000|121801|(2<sup>661478</sup>+1)<sup>2</sup>-2}}||398250||[[Mark Rodenkirch]]||2016-06-18
 +
|-
 +
|(1968<sup>58533</sup>+1)<sup>2</sup>-2||385619||[[Clint Stillman]]||2017-11-30
 +
|-
 +
|(2<sup>621443</sup>+1)<sup>2</sup>-2||374146||[[Mark Rodenkirch]]||2016-05-30
 +
|}
  
 
==Data==
 
==Data==

Revision as of 08:12, 19 June 2019

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

Top 5 Carol primes

Prime Digits Found by Date
(290124116-1)2-2 611246 Karsten Bonath 2019-03-01
(2695631-1)2-2 418812 Mark Rodenkirch 2016-07-16
(2688042-1)2-2 414243 Mark Rodenkirch 2016-07-05
(17887525-1)2-2 393937 Serge Batalov 2016-05-21
(2653490-1)2-2 393441 Mark Rodenkirch 2016-06-03

Top 5 Kynea primes

Prime Digits Found by Date
(362133647+1)2-2 683928 Karsten Bonath 2019-06-17
(30157950+1)2-2 466623 Serge Batalov 2016-05-22
(2661478+1)2-2 398250 Mark Rodenkirch 2016-06-18
(196858533+1)2-2 385619 Clint Stillman 2017-11-30
(2621443+1)2-2 374146 Mark Rodenkirch 2016-05-30

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

Number classes
General numbers
Special numbers
Prime numbers
Retrieved from "https://www.rieselprime.de/z/index.php?title=Carol-Kynea_prime&oldid=3145"