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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]\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).


Carol and Kynea numbers were first studied by Cletus Emmanuel in 1995[1], who named them after personal acquaintances[2]. He searched these forms for primes up to the limit of 15000.

Starting in 2004, Steven Harvey maintained a search for this form. At this time Multisieve and cksieve 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 2015-12-26 Mark opened a thread[3] 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
(2852770+1)2-2 513419 Ryan Propper 2019-07-14
(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

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 381 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 62 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.

How to participate?


  • Reserve your base(s)/range(s) in this thread.


  • Use cksieve (from Mtsieve) and
    • run a new sieve by calling cksieve -b 12 -n 1 -N 10000 -P 1000000000 (for base=12, n-range=1-10000, max prime factor 109). The sieve file will be written to ck_12.pfgw.
    • rerun an old sieve by calling cksieve -P 1e12 -i ck_12.pfgw -o ck_12.pfgw -f factors.txt (for base=12, max prime factor 1012, input/output files given, storing factors to "factors.txt").

PRP testing

  • Use PFGW calling pfgw64.exe -f0 ck_12.pfgw (running candidates file for base 12, no further factoring).

Prime testing

After testing with PFGW higher probable primes will be written in "pfgw.log". These have to be checked prime by calling like pfgw64 -tp -q"(12^68835-1)^2-2".


Once you have completed your range, report any primes found in this thread. Then report the completed range in the reservation thread and specify whether you will continue with the base or release it.


External links



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General numbers
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Prime numbers