From tonyforbes@ltkz.demon.co.uk Sat Dec  6 01:41:52 1997
Date: Fri, 5 Dec 1997 18:07:55 -0500
From: Tony Forbes <tonyforbes@ltkz.demon.co.uk>
To: NMBRTHRY@LISTSERV.NODAK.EDU
Subject: Cunningham Chain of length 16

Cunningham Chain of length 16
-----------------------------

Tony Forbes
-----------

A Cunningham chain of length k is a finite set of primes {p_1, p_2, ...,
p_k}, where either

   p_{i+1} = 2*p_i + 1,  i = 1, 2, ..., k - 1,

or

   p_{i+1} = 2*p_i - 1,  i = 1, 2, ..., k - 1.

The subject is discussed in Section A7 of Guy's book [1] in which it is
reported that Gunter Loh [2] discovered two 12-chains of the first type
and one 13-chain of the second type. The author [3, 4] has reported
several 14-chains of both types and one 15-chain of the second type.

On 5 December 1997 the author discovered a Cunningham chain with 16
elements.

The program uses a 16-way sieving procedure on a system of 32 parallel
tables of words (one table for each bit). On a Pentium this is faster
than using a single table of bits. Because the sieving primes are
relatively small, the cost of computing [x/32] and 2^(x mod p) is
significantly greater than the extra work required for setting up 32
small tables.

The Cunninghan chain is of the second type, and here it is.

   {3203000719597029781,
    6406001439194059561,
    12812002878388119121,
    25624005756776238241,
    51248011513552476481,
    102496023027104952961,
    204992046054209905921,
    409984092108419811841,
    819968184216839623681,
    1639936368433679247361,
    3279872736867358494721,
    6559745473734716989441,
    13119490947469433978881,
    26238981894938867957761,
    52477963789877735915521,
    104955927579755471831041}.

References

[1] Richard K. Guy, Unsolved Problems in Number Theory, Springer Verlag,
Berlin, Second edition, 1994.

[2] Gunter Loh, Long chains of nearly doubled primes, Math. Comp., 53
(1989) 751-759.

[3] Tony Forbes, Cunningham Chains of length 14, NMBRTHRY, May 1997.

[4] Tony Forbes, Cunningham Chain of length 15, NMBRTHRY, September
1997.

========================================================================


Harvey Dubner, Paul Zimmermann and the author are currently organizing a
search for 9 consecutive primes in arithmetical progression. We are
still seeking helpers; if you would like to participate, please visit
one of our Web sites.

For workstations, or PC's under Linux:

   http://www.loria.fr/~zimmerma/records/9primes.html

For PC's under Windows:

   http://www.ltkz.demon.co.uk/ar2/9primes.htm

--
Tony