Cunningham Chains (2nd kind)

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

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

where

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

Record Primes of this Type

rank prime digits who when comment

1 3853775193 · 2⁸⁰⁰⁰¹+1 24093 L109 May 2007 Cunningham chain 2nd kind (2p-1)
2 1504084599 · 2⁷⁸³⁴²+1 23593 g290 Apr 2004 Cunningham chain 2nd kind (2p-1)
3 964487139 · 2⁷⁸³⁴²+1 23593 g290 Apr 2004 Cunningham chain 2nd kind (2p-1)
4 787302705 · 2⁵⁶⁷⁹⁰+1 17105 g336 Mar 2005 Cunningham chain 2nd kind (2p-1)
5 40931485 · 2⁵³¹²⁴-3 16000 p222 Mar 2007 Cunningham chain 2nd kind (2p-1)
6 2366867925 · 2¹⁷²⁰⁸+1 5190 p133 Oct 2004 Cunningham chain 2nd kind (4p-3)
7 1793349831 · 2¹⁵²⁵⁷+1 4603 p133 Jul 2004 Cunningham chain 2nd kind (4p-3)
8 1110159213 · 2¹⁵¹⁶⁶+1 4575 g250 Oct 2002 Cunningham chain 2nd kind (4p-3)
9 3345660375 · 2¹⁵¹²⁷+1 4564 p94 Oct 2002 Cunningham chain 2nd kind (4p-3)
10 1531785651 · 2¹⁰¹⁰⁹+1 3053 g250 Aug 2001 Cunningham chain 2nd kind (4p-3)
11 11628008104 · 4127#+1 1770 p133 Mar 2005 Cunningham chain 2nd kind (8p-7)
12 1226756544 · 4001#+1 1712 p133 Apr 2004 Cunningham chain 2nd kind (8p-7)
13 1054831232256 · 3061#+1 1314 p44 Jan 2003 Cunningham chain 2nd kind (8p-7)
14 2853609856 · 3041#+1 1304 p94 Oct 2002 Cunningham chain 2nd kind (8p-7)
15 2591184354165 · 2³²⁹⁰+1 1003 p151 Nov 2004 Cunningham chain 2nd kind (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 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 / B_k.

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

rank prime digits who when comment

1 11628008104 · 4127#+1 1770 p133 Mar 2005 Cunningham chain 2nd kind (8p-7)
2 1226756544 · 4001#+1 1712 p133 Apr 2004 Cunningham chain 2nd kind (8p-7)
3 1054831232256 · 3061#+1 1314 p44 Jan 2003 Cunningham chain 2nd kind (8p-7)
4 2853609856 · 3041#+1 1304 p94 Oct 2002 Cunningham chain 2nd kind (8p-7)
5 2591184354165 · 2³²⁹⁰+1 1003 p151 Nov 2004 Cunningham chain 2nd kind (8p-7)
6 2366867925 · 2¹⁷²⁰⁸+1 5190 p133 Oct 2004 Cunningham chain 2nd kind (4p-3)
7 1793349831 · 2¹⁵²⁵⁷+1 4603 p133 Jul 2004 Cunningham chain 2nd kind (4p-3)
8 1110159213 · 2¹⁵¹⁶⁶+1 4575 g250 Oct 2002 Cunningham chain 2nd kind (4p-3)
9 3345660375 · 2¹⁵¹²⁷+1 4564 p94 Oct 2002 Cunningham chain 2nd kind (4p-3)
10 1531785651 · 2¹⁰¹⁰⁹+1 3053 g250 Aug 2001 Cunningham chain 2nd kind (4p-3)
11 3853775193 · 2⁸⁰⁰⁰¹+1 24093 L109 May 2007 Cunningham chain 2nd kind (2p-1)
12 1504084599 · 2⁷⁸³⁴²+1 23593 g290 Apr 2004 Cunningham chain 2nd kind (2p-1)
13 964487139 · 2⁷⁸³⁴²+1 23593 g290 Apr 2004 Cunningham chain 2nd kind (2p-1)
14 787302705 · 2⁵⁶⁷⁹⁰+1 17105 g336 Mar 2005 Cunningham chain 2nd kind (2p-1)
15 40931485 · 2⁵³¹²⁴-3 16000 p222 Mar 2007 Cunningham chain 2nd kind (2p-1)

Dirk Augustin's Cunningham Chain Records
The Top 20's Cunningham Chains (first kind)
The Prime Links'++ list of Cunningham Chain links
The Prime Glossary's Cunningham Chain Entry

References