- 1 Definitions
- 2 History
- 3 Top 5 Carol primes
- 4 Top 5 Kynea primes
- 5 OEIS sequences
- 6 Data
- 7 How to participate?
- 8 References
- 9 External links
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).
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 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
Top 5 Kynea primes
These are available OEIS sequences:
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 84 sequences.
Bases without a Kynea prime
There are 75 sequences.
Bases without a Carol and Kynea prime
There are 2 sequences.
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").
- run a new sieve by calling
- Use PFGW calling
pfgw64.exe -f0 ck_12.pfgw(running candidates file for base 12, no further factoring).
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.