Prime Constellation Records

Primary record pages: tuplets (external), AP's, CPAP's, CC's, simultaneous, gaps, BiTwins (external)

This page links maintained record pages for prime constellations, here meaning sets of at least 2 primes in some pattern.
Pages with other information about prime constellations are usually not linked here but maybe at MathWorld: Prime Clusters.

My comments to the links are in parentheses below.
"primary": I consider it the primary record page for that category (I made some pages and may be biased).
"top-N": The N largest constellations are listed, usually for each number of primes.
"only titanic": Only primes with at least 1000 digits are listed.

Here a prime k-tuplet means k primes as closely together as possible for primes above k.
Tony Forbes: Prime k-tuplets (primary, top-10, also smallest)
Tony Forbes: List of all possible patterns of prime k-tuplets (up to k=50)
Chris Caldwell: The Top Twenty: Twin Primes (top-20, only twins)
Chris Caldwell: Database tuplet search (top-5, only titanic, not twins)
Thomas J Engelsma: K-Tuple Permissible Patterns (only about patterns, no primes)
Matroids Matheplanet Forum: smallest prime k-tuple - for several k and each digit (forum thread in German and English)

Arithmetic progressions
An AP-k is k primes in arithmetic progression: pn = p0  + d n, for n = 0 ... k-1.
A CPAP-k is k consecutive primes in AP, i.e. there are no other primes from p0 to pk-1.
Jens K Andersen: Primes in Arithmetic Progression Records (primary, top-1, also smallest start, end, difference)
Jens K Andersen: The Largest Known CPAP's (primary, top-10, also smallest CPAP and largest difference)
Chris Caldwell: The Top Twenty: Arithmetic Progressions of Primes (mixed list, only titanic, also weighted top-20 but not based on computational difficulty)
Chris Caldwell: The Top Twenty: Consecutive Primes in Arithmetic Progression (top-5, only titanic)
Chris Caldwell: Database search for last prime in AP-3, -4, -5, -6, -7, -8 (many AP-3, top-5 of the others, only titanic).

Cunningham Chains
A CC-k is a Cunningham Chain of length k, i.e. k primes where each is twice the previous, plus one (CC of the 1st kind) or minus one (CC of the 2nd kind). E.g. a CC-3 of the 2nd kind is (p, 2p-1, 4p-3). Sophie Germain primes is a CC-2 of the first kind: (p, 2p+1).
Dirk Augustin: Cunningham Chain records (primary, top-3, also smallest)
Chris Caldwell: The Top Twenty: Sophie Germain (p) (top-20)
Chris Caldwell: The Top Twenty: Cunningham Chain (1st kind) (top-5, not Sophie Germain, only titanic, also weighted top-20 but not based on computational difficulty)
Chris Caldwell: The Top Twenty: Cunningham Chain (2nd kind) (top-5, only titanic, also weighted top-20 but not based on computational difficulty)

Prime gaps
A prime gap is an interval of consecutive composite numbers between 2 primes (probable primes sometimes accepted). Some would not call it a prime constellation.
Jens K Andersen: The Top-20 Prime Gaps (primary, top-20 for absolute and relative size, plus some combinations)
Thomas R Nicely: First occurrence prime gaps (smallest existing or known primes for each gap size, tables of prime gaps)
Chris Caldwell: The Gaps Between Primes (theorems and conjectures about gap sizes)

Other prime constellations
Henri Lifchitz: BiTwin records (up to top-20, also generalized BiTwins, description of BiTwins)
A BiTwin with n links is n+1 twin prime pairs k 2^m +/- 1 for constant k and n+1 consecutive m values.
A generalized BiTwin replaces "2" with a larger constant.

Jens K Andersen: The Largest Known Simultaneous Primes (top-1, largest overall case of "simultaneous" primes)

Carlos Rivera: The Prime Puzzles & Problems Connection (many of the puzzles and problems involve prime constellations, e.g. puzzle 20, 54, 122, 167, 181, 206, 255, 350)

There are also some constellation related sequences (among 100000+ other sequences!) in the OEIS:
N. J. A. Sloane: The On-Line Encyclopedia of Integer Sequences

If you know a sequence of small numbers or one larger number related to a prime constellation, you can try to search the numbers at the OEIS or Google. I have sometimes found previous work this way.

Jens K Andersen: My prime records (link collection to my own records)

Page made by Jens Kruse Andersen,    home  
Please mail if you know a link to a maintained prime constellation record page.
Last updated 27 March 2018.