Newsgroups: sci.math
Subject: Prime constellations.
From: wft@math.canterbury.ac.nz (Bill Taylor)
Date: 27 Aug 92 18:51:55 +1200
Distribution: world
Organization: Department of Mathematics, University of Canterbury

Some time ago, there was a brief discussion concerning
"prime constellations", that is , a set of N integers
 ~~~~~~~~~~~~~~~~~~~~       { 0, a2, a3, ..., aN }    such that

{ p, p+a2, p+a3, ... p+aN } might all be prime, for infinitely many p.
So that for instance, the TWIN PRIME conjecture concerns constellation { 0, 2 } .

The suggestion was made that constellations might have solutions in
primes, for either infinitely many primes "p", or only 0 or 1 such "p".

That infinitely-many-primes cases exist, is a famous conjecture,
connected with Hardy, Bateman and others.

The suggestion that the only finitely-many-solutions could be achieved for
cases 0 and 1 was soon shot down by many examples,
e.g. the constellation A = { 0, 2, 8, 14, 26 } has 2 solutions

A+3 = { 3, 5, 11, 17, 29 }  and
A+5 = { 5, 7, 13, 19, 31},

and as A includes all residues [mod 5] there cannot be any further solutions.

Chris Long outlined a nice little ad hoc procedure for generating such
2-solution constellations, and asked for the smallest (in some sense)
constellation that has exactly three solutions.
                       ~~~~~~~~~~~~~~~~~~~~~~~
There seem to have been no further followups.  A pity.

To try and resuscitate interest I decided to post my rather boring solution.

I extended Chris Long's method to generate a 3-solution constellation.
The smallest possible pre-constellation pattern I could find was

 7 13 37 ...     OR      7 31 37 ...
31 37 61 ...            13 37 43 ...
37 43 67 ...            37 61 67 ...

Either of these can be the start of 3 solutions for a constellation of
cardinality 37, and, provided the 37 entries (on the top line, say)
include every possible number [mod 37], there can be no further solution.
Thus we would get a constellation with *exactly* 3 solutions.

As far as I could see, 37 is the smallest possible integer with this property.
                       ~~~~~~~~~~~~~~~~~~
So, I then set Maple to work to find the first 34 extra numbers that would
give all primes, in all 3 rows, and cover all 37 residues [mod 37].

It found that the left-hand pattern gave the smallest numbers, giving
a constellation A with
                         A+7 =

{ 7, 13, 37, 43, 59, 73, 79, 83, 107, 127, 149, 167, 199, 233, 239, 283, 359,
 409, 419, 479, 547, 563, 617, 797, 829, 953, 1009, 1039, 1063, 1399, 1429, 1669,
 1723, 1847, 2659, 3583, 3607 }

These, and A+31, and A+37  give all primes, and no other A+p can do so.

And this is (I think) in sense above, the smallest such example.
----
Well; I warned you it would be pretty boring !  There is no problem that
I can see with extending the method to get 4-fold solutions, but the numbers
involved are going to be pretty horrific, I would say.

It would be nice to see some further action on this thread !

Cheers,
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            Bill Taylor              wft@math.canterbury.ac.nz
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Newsgroups: sci.math