A020497

[David W. Wilson]

Olivier Gerard wrote:

> Dear David,
>
> Could you post a definition of "constellation"
> to seqfan, so that members could more easily
> follow the thread ?
>
> regards,
>
> Olivier

 I was unable to find a definition of constellation, even in Erics's
Encyclopedia, though the term is in common usage.

Disclaimer:  I do not understand all the complexities of pattern
theory.  Here I am taking a very simplistic approach.

Let S be a finite set of integers, and let m be the least element of
S, and let C = S-m = { s-m | s in S }.  We say that S forms
constellation C.

For example, the set S = { 5, 7, 12 } forms the constellation
{ 5, 7, 12 } - 5 = { 0, 2, 7 }.  Note that (in this oversimplified
development) every constellation has least element 0.

C is called a prime constellation if it is the constellation of a set
of primes.  C = { 0, 2, 4 } is a prime constellation, being the
constellation of { 3, 5, 7 }.  { 0, 1, 2 } is not a prime constellation
since there are no three consecutive integers that are prime.

A constellation C is called admissible if C does not contain all the
residues of any prime.  For instance, C = { 0, 2, 4 } is not
admissible, because it contains all the residues of 3.  On the other
hand, C = { 0, 2, 6 } is admissible.

It can be shown that if C is not admissible, there are only a finite
number of sets of primes having constellation C.  For instance,
C = { 0, 1, 2 } is not admissible (it contains all residues of 2), and
therefore C is the constellation of only a finite number of sets of
primes (in fact, none at all).  Similarly, C = { 0, 2, 4 } is not
admissible, and is the constellation of only one set of primes, namely
S = { 3, 5, 7 }.

It is conjectured that the converse is also true, specifically, that
if C is admissible, it is the constellation of an infinite number of
sets of primes.  (This is a corollary of the k-tuple conjecture; see
http://www.astro.virginia.edu/~eww6n/math/k-TupleConjecture.html.
There k-tuple conjecture has many other interesting corollaries, e.g,
the existence of arbitrarily long arithmetic progressions of primes).

Assuming the k-tuple conjecture, the admissibility of C = { 0 }
implies that there are an infinite number of prime singletons { p },
that is, an infinite number of primes, which we know to be true.
Similarly, the admissibility of C = { 0, 2 } leads to conclude that
there are an infinite number of prime pairs { p, p+2 }; this is the
twin primes conjecture, still unproved, but almost certainly true.
The admissibility of C = { 0, 2, 6 } and C = { 0, 4, 6 } implies that
there are an infinitude of prime triples of the form { p, p+2, p+6 }
and { p, p+4, p+6 }, also credible.