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.

**Tuplets**

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: p_{n} = p_{0}
+ d · n, for n = 0 ... k-1.

A CPAP-k is k *consecutive* primes in AP, i.e. there are no other primes
from p_{0} to p_{k-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,
jens.k.a@get2net.dk home

Please mail if you know a link to a maintained prime constellation record page.

Last updated 27 March 2018.