From: Paul.Jobling@WhiteCross.com (Paul Jobling)
Subject: Arithmetic progressions of twin primes
Date: 17 Jan 00 14:17:11 GMT
Newsgroups: sci.math.numberthy
Summary: [missing]

I have been investigating arithmetic progressions of twin primes, where
a twin AP of length n consists of 2n primes,

a +-1, a+b +-1, a+2b +-1, ..., a+(n-1)b +-1.

These are similar to sequences of primes in AP, for which Grosswald has
an estimate of the number of sequences of length n in the primes up to
some limit x. In particular, this estimate indicated that there are
sequences of arbitrary length. This estimate was experimentally checked
by Pritchard et al, who managed to find an AP of 22 primes.

For twin primes in AP, I do not know of a similar published estimate. I
conjecture, however, that for twin primes it is _not_ the case that
sequences of arbitrary length can be found.

For an AP of twin primes of length n, it is obvious that b must be a
multiple of n#, the product of the primes <=n. I therefore performed the
following search:

For p= the primes 11, 13, 17:
  For each possible x <= p#
    Obtain the 1 million numbers of the form x + i.p#, 0 <= i < 1000000
    Discard those i's where x+i.p#+1 or x+i.p#-1 is composite.
    This will leave those i's where x+i.p# +- 1 is a pair of twin
primes.
    Search these i's for an arithmetic progression of length >= 8.

Note that it seems to be quite a difficult problem to find an efficient
algorithm to detect an AP in a list of numbers.

The testing time was very short for p=11 and 13 (hours); for p=17 the
search time was approximately 40 days on a 233Mhz PII.

I had hoped to find an AP of 11 twin primes, consisting of 22 primes.
Unfortunately the best I could manage were the following APs of 10 twin
primes, 20 primes in all, listed in increasing order of their final
term:

i=0 to 9:

(7146+i*7087)*17#+239670 +- 1
(27193+i*15352)*17#+39930 +- 1
(103299+i*8702)*17#+409602 +- 1
(240056+i*1185)*17#+399000 +- 1
(28070+i*24909)*17#+369168 +- 1
(43711+i*33725)*17#+160878 +- 1
(392688+i*2040)*17#+429018 +- 1
(263377+i*18202)*17#+359172 +- 1
(1952+i*70022)*17#+392772 +- 1
(104521+i*62645)*17#+353532 +- 1
(189110+i*89957)*17#+244200 +- 1
(419358+i*62769)*17#+424980 +- 1


Regards,

Paul.