Cunningham Chain of length 16
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Tony Forbes
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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 Cunninghan 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.