Carol-Kynea prime

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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).


Carol and Kynea numbers were first studied by Cletus Emmanuel, who named them after personal acquaintances. He searched these forms for primes up to the limit of 15000. Starting in 2004, S. Harvey maintained a search for this form. At this time Multisieve and ck were used to sieve these forms and PFGW was used to test for primality. The search went dormant in 2011 and was resurrected in 2015 by Mark Rodenkirch. Initially Multisieve was used, but then later on he wrote cksieve which would later become part of mtsieve framework. On April 15, 2016 Mark opened a thread for a coordinated search of Carol/Kynea numbers on Mersenneforum, which continues to this day (although now Gary Barnes, maintainer of NPLB and CRUS, maintains the search).

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

OEIS sequences

These are available OEIS sequences:

Base Carol Kynea
2 A091515 A091513
6 A100901 A100902
10 A100903 A100904
14 A100905 A100906
22 A100907 A100908


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

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General numbers
Special numbers
Prime numbers