The Top Twenty--a Prime Page Collection

Cunningham Chains (1st kind)

This page : Definition(s) | Records | Weighted Records | References | Related Pages |

The Prime Pages

The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page. This page is about one of those forms. Comments and suggestions requested.

Definitions and Notes

Recall that a Sophie Germain prime is a prime p such that q=2p+1 is also prime. Why not also ask that r=2q+1 is prime, and 2r+1 is prime, and...? A Cunningham chain of length k of the first kind is a sequence of k primes, each which is twice the proceeding one plus one. For example, {2, 5, 11, 23, 47} and {89, 179, 359, 719, 1439, 2879}.

We have a separate page about the Cunninham chains of the second kind and the Sophie Germain primes. Cunningham chains (of both kinds) are also called chains of nearly doubled primes.

For any given length k there should be infinitely many chains of length k. In fact the number less than N should be asymptotic to

where

where the sequence B_k begins approximately 1.32032 (k=2), 2.85825, 5.553491, 20.2636, 71.9622, 233.878, 677.356.

Because the Sophie Germain primes have their own Top Twenty page, they are not included in the list of records below. Forthe same reason 'Cunningham Chain (p)'s and 'Cunningham Chain (2p+1)'s (which are Sophie Germain primes) are omitted.

Record Primes of this Type

rank prime digits who when comment

1 379185609 · 2²⁷¹²⁹-1 8176 L983 Nov 2009 Cunningham chain (4p+3)
2 164210699973 · 2²⁶³²⁸-1 7937 p158 Aug 2006 Cunningham chain (4p+3)
3 3020255265 · 2²⁰⁰²⁵-1 6038 p133 Apr 2005 Cunningham chain (4p+3)
4 16219299585 · 2¹⁶⁶¹⁴-1 5012 p158 Jan 2005 Cunningham chain (4p+3)
5 2288999415 · 2¹⁵⁹³⁹-1 4808 p133 Jul 2004 Cunningham chain (4p+3)
6 953477584 · 5501#-1 2355 p133 Jun 2005 Cunningham chain (8p+7)
7 41812496896 · 3067#-1 1316 p133 Aug 2004 Cunningham chain (8p+7)
8 191881920 · 3067#-1 1314 p133 Jul 2004 Cunningham chain (8p+7)
9 13657785480 · 3049#-1 1309 p94 Oct 2002 Cunningham chain (8p+7)
10 19743490208 · 2801#-1 1208 p133 Mar 2005 Cunningham chain (8p+7)

Weighted Record Primes of this Type

For amusement purposes only we might seek to weight the chains on the list of largest known primes by an estimate of how rare chains of that length are. We also include the Sophie Germain primes because they are chains of length two.

To form a weight we start with the usual estimate of how hard it is to prove primality of a number the size of n

log(n)² log log n

and multiply it by the expected number of potential candidates to test before we find one of length k (by the heuristic estimate above)

log(n)^k / B_k.

We then take the log one more time to make the numbers nice and small.

(Because the Sophie germain primes have their own Top Twenty page, they are not included in the list of records below.)

rank prime digits who when comment

1 953477584 · 5501#-1 2355 p133 Jun 2005 Cunningham chain (8p+7)
2 379185609 · 2²⁷¹²⁹-1 8176 L983 Nov 2009 Cunningham chain (4p+3)
3 164210699973 · 2²⁶³²⁸-1 7937 p158 Aug 2006 Cunningham chain (4p+3)
4 41812496896 · 3067#-1 1316 p133 Aug 2004 Cunningham chain (8p+7)
5 191881920 · 3067#-1 1314 p133 Jul 2004 Cunningham chain (8p+7)
6 13657785480 · 3049#-1 1309 p94 Oct 2002 Cunningham chain (8p+7)
7 19743490208 · 2801#-1 1208 p133 Mar 2005 Cunningham chain (8p+7)
8 3020255265 · 2²⁰⁰²⁵-1 6038 p133 Apr 2005 Cunningham chain (4p+3)
9 16219299585 · 2¹⁶⁶¹⁴-1 5012 p158 Jan 2005 Cunningham chain (4p+3)
10 2288999415 · 2¹⁵⁹³⁹-1 4808 p133 Jul 2004 Cunningham chain (4p+3)

Related Pages

Dirk Augustin's Cunningham Chain Records
The Top 20 Cunningham Chains of the Second Kind
The Prime Links++ list of Cunningham Chain links
The Cunningham Chain's entry from the Prime Glossary

References

Cunningham1907
A. Cunnningham, "On hyper-even numbers and on Fermat's numbers," Proc. Lond. Math. Soc., series 2, 5 (1907) 237--274.
Guy94 (SectionA7)
R. K. Guy, Unsolved problems in number theory, Springer-Verlag, New York, NY, 1994. ISBN 0-387-94289-0. MR 96e:11002 [An excellent resource! Guy briefly describes many open questions, then provides numerous references.]
Lehmer1965
D. H. Lehmer, "On certain chains of primes," Proc. Lond. Math. Soc., series 3, 14a (1965) 183--186. MR 31:2222
LM1980
C. Lalout and J. Meeus, "Nearly-doubled primes," J. Recreational Math., 13 (1980-81) 30--35.
Loh89
G. Löh, "Long chains of nearly doubled primes," Math. Comp., 53 (1989) 751-759. MR 90e:11015 (Abstract available) (Annotation available)
Ribenboim95 (p 333)
P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995. pp. xxiv+541, ISBN 0-387-94457-5. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]
Yates82
S. Yates, Repunits and repetends, Star Publishing Co., Inc., Boynton Beach, Florida, 1982. pp. vi+215, MR 83k:10014

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