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 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.