prime constellation

Prime constellations of length k are the shortest admissible k-tuples of primes. That is, a k-tuple is admissible unless there is a prime q ≤ k, which always divides the product of the terms. Below we illustrate this tricky definition with a couple examples.

Suppose we want to find prime constellations of length two. The pattern (p, p+1) is not admissible, because either p or p+1 would be even, so one of these would have to be the prime 2. The pattern (p, p+2) is admissible (both p and p+2 can be odd), so prime constellations of length two fit the pattern (p, p+2). Examples: (3,5), (5,7), (11,13), (17,19), and (29,31). Of course, we call primes that fit this pattern twin primes.

Now let us find a prime constellation of length three. The primes in our pattern must differ by at least two (so we are not forcing one of them to be two). So we might try (p, p+2, p+4). But one of these three must be divisible by 3, so this also is not admissible. Finally, we consider

Both of these are admissible, so all prime constellations of length three have one of these forms. (Examples: (5,7,11), (7,11,13), (11,13,17), (13,17,19) and (17,19,23).) These are prime triples.

Prime constellations of length four (prime quadruplets) must fit the single pattern (p, p+2, p+6, p+8). (Examples: (5,7,11,13), (11,13,17,19)). If the length is five or six we have the patterns: (p, p+2, p+6, p+8, p+12), (p, p+4, p+6, p+10, p+12), and (p, p+4, p+6, p+10, p+12, p+16). It is conjectured that there are infinitely many of each admissible prime constellation (see the prime k-tuple conjecture). The first link below contains more information, including lists of admissible patterns of larger lengths, as well as the largest known prime constellations of lengths 2, 3, 4, ..., 17.

See Also: PrimeKTuplet

Related pages (outside of this work)

References:

Forbes1999
T. Forbes, "Prime clusters and Cunningham chains," Math. Comp., 68:228 (1999) 1739--1747.  MR 99m:11007
Guy94 (A9)
R. K. Guy, Unsolved problems in number theory, Springer-Verlag, 1994.  New York, NY, ISBN 0-387-94289-0. MR 96e:11002 [An excellent resource! Guy briefly describes many open questions, then provides numerous references. See his newer editions of this text.]
Ribenboim95
P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995.  pp. xxiv+541, ISBN 0-387-94457-5. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]
Riesel94 (Chapter Three)
H. Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics Vol, 126, Birkhäuser Boston, Boston, MA, 1994.  ISBN 0-8176-3743-5. MR 95h:11142 [An excellent reference for those who want to start to program some of these algorithms. Code is provided in Pascal. Previous edition was vol. 57, 1985.]
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