The Top Twenty--a Prime Page Collection

Cunningham Chains (2nd kind)

This page : Definition(s) | Records | Weighted Records | References | Related Pages | RSS 2.0 Feed
  View this page in:   language help

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.

(up) Definitions and Notes

A Cunningham chain of length k of the second kind is a sequence of k primes, each which is twice the proceeding one minus one. (For example, {2, 3, 5} and {1531, 3061, 6121, 12241, 24481}.)

We have a separate page about Cunningham chains of the first kind. 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

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

(up) Record Primes of this Type

rankprime digitswhowhencomment
1648309 · 2148311 + 1 44652 L983 Jul 2010 Cunningham chain 2nd kind (2p - 1)
2552903 · 2136157 + 1 40994 L983 Jun 2010 Cunningham chain 2nd kind (2p - 1)
3163221 · 2124601 + 1 37514 L983 Dec 2009 Cunningham chain 2nd kind (2p - 1)
42243973027 · 2104568 + 1 31488 L99 Mar 2012 Cunningham chain 2nd kind (2p - 1)
5532323 · 2104390 + 1 31431 L983 Nov 2009 Cunningham chain 2nd kind (2p - 1)
682659189 · 226999 + 1 8136 L983 Mar 2010 Cunningham chain 2nd kind (4p - 3)
7173028555 · 226995 + 1 8135 L983 Feb 2010 Cunningham chain 2nd kind (4p - 3)
842989535 · 226545 + 1 7999 L983 Mar 2010 Cunningham chain 2nd kind (4p - 3)
955339803 · 219402 + 1 5849 L983 Dec 2009 Cunningham chain 2nd kind (4p - 3)
10387977793 · 217866 + 1 5387 L983 Oct 2009 Cunningham chain 2nd kind (4p - 3)
115045589688 · 4933# + 1 2106 p295 Dec 2010 Cunningham chain 2nd kind (8p - 7)
12125848198864 · 4253# + 1 1829 p199 Nov 2010 Cunningham chain 2nd kind (8p - 7)
13113419228920 · 4253# + 1 1829 p199 Nov 2010 Cunningham chain 2nd kind (8p - 7)
1445912427272 · 4253# + 1 1829 p199 Nov 2010 Cunningham chain 2nd kind (8p - 7)
1511628008104 · 4127# + 1 1770 p133 Mar 2005 Cunningham chain 2nd kind (8p - 7)
161290733709840 · 2677# + 1 1141 p295 Jan 2011 Cunningham chain 2nd kind (16p - 15)
17720128166480 · 2621# + 1 1117 p199 Jul 2010 Cunningham chain 2nd kind (16p - 15)
18418059269664 · 2371# + 1 1015 p308 Apr 2011 Cunningham chain 2nd kind (16p - 15)

(up) 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 might 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 / Bk.
We then take the log one more time to make the numbers nice and small.

rankprime digitswhowhencomment
11290733709840 · 2677# + 1 1141 p295 Jan 2011 Cunningham chain 2nd kind (16p - 15)
2720128166480 · 2621# + 1 1117 p199 Jul 2010 Cunningham chain 2nd kind (16p - 15)
3418059269664 · 2371# + 1 1015 p308 Apr 2011 Cunningham chain 2nd kind (16p - 15)
45045589688 · 4933# + 1 2106 p295 Dec 2010 Cunningham chain 2nd kind (8p - 7)
5125848198864 · 4253# + 1 1829 p199 Nov 2010 Cunningham chain 2nd kind (8p - 7)
6113419228920 · 4253# + 1 1829 p199 Nov 2010 Cunningham chain 2nd kind (8p - 7)
745912427272 · 4253# + 1 1829 p199 Nov 2010 Cunningham chain 2nd kind (8p - 7)
811628008104 · 4127# + 1 1770 p133 Mar 2005 Cunningham chain 2nd kind (8p - 7)
982659189 · 226999 + 1 8136 L983 Mar 2010 Cunningham chain 2nd kind (4p - 3)
10173028555 · 226995 + 1 8135 L983 Feb 2010 Cunningham chain 2nd kind (4p - 3)
1142989535 · 226545 + 1 7999 L983 Mar 2010 Cunningham chain 2nd kind (4p - 3)
1255339803 · 219402 + 1 5849 L983 Dec 2009 Cunningham chain 2nd kind (4p - 3)
13387977793 · 217866 + 1 5387 L983 Oct 2009 Cunningham chain 2nd kind (4p - 3)
14648309 · 2148311 + 1 44652 L983 Jul 2010 Cunningham chain 2nd kind (2p - 1)
15552903 · 2136157 + 1 40994 L983 Jun 2010 Cunningham chain 2nd kind (2p - 1)
16163221 · 2124601 + 1 37514 L983 Dec 2009 Cunningham chain 2nd kind (2p - 1)
172243973027 · 2104568 + 1 31488 L99 Mar 2012 Cunningham chain 2nd kind (2p - 1)
18532323 · 2104390 + 1 31431 L983 Nov 2009 Cunningham chain 2nd kind (2p - 1)

(up) Related Pages

(up) 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. See his newer editions of this text.]
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
Chris K. Caldwell © 1996-2012 (all rights reserved)