20070814, 10:20  #1 
Jun 2003
Oxford, UK
11×173 Posts 
Generalised Cunningham Chains
A Cunningham Chain length 2 is k*2^n+/1, where n= x and x+1 both produce primes. Longer chains can be created of length y when n=x to x+(y1) all produce primes.
Not much has been done in exploring other bases other than 2, which are a sub set of Generalised Cunningham Chains (GCC). Here are some early GCC of at least length 9 in various bases: 10347747270980*3^n+1, n from 1 to 10 all prime = GCC(3)10 550326588*5^n+1, n from 1 to 10 = GCC(5)10 678979904460*7^n+1, n from 1 to 9 = GCC(7)9 943151976*11^n+1, n from 1 to 9 = GCC(11)9 2924027880*23^n+1, n from 1 to 9 = GCC(23)9 91636690860*23^n+1, n from 1 to 9 = GCC(23)9 Please post to this thread any improvements or new GCC(base)9+ at each base level. Please note base does not have to be a prime. Base 10 is of interest to the repdigit gangs. Regards Robert Smith Last fiddled with by robert44444uk on 20070814 at 10:23 
20070815, 07:14  #2 
Jun 2003
Oxford, UK
1903_{10} Posts 
GCC(4)11!
95472622*4^n+1, n from 1 to 11 is a GCC(4)11!
Here are some GCC(4)10 's for "+1" and from n=1 to 10 k= 261716590 805489743 972653203 Last fiddled with by robert44444uk on 20070815 at 07:17 
20070816, 08:48  #3 
Jun 2003
Oxford, UK
76F_{16} Posts 
GCC(4)13!!!!!
Found my first chain of 13,
6703351518*4^n1, n from 1 to 13 all prime!! 
20070822, 06:46  #4 
Jun 2003
Oxford, UK
11·173 Posts 
Using the notation GCC(base, + or )"length of Generalised Cunningham Chain" for the form k*base^n+/1, Some GCC(9,+)9 k values:
k= 1081477811 1283151520 1468201379 4156073600 3920061569 3791210290 3715720289 4912720955 4441618689 Last fiddled with by robert44444uk on 20070822 at 07:05 
20070822, 14:04  #5  
"William"
May 2003
New Haven
2^{2}·3^{2}·5·13 Posts 
Quote:
1081477811*9^{n}+1 is always even. What is the correct expression for the primes? 

20070824, 14:32  #6 
Jun 2003
Oxford, UK
11×173 Posts 

20070825, 04:25  #7 
Jun 2003
Oxford, UK
11×173 Posts 
k=84378963 is GCC(16,+)11
Bases that are squares are particularly rich as ModuloOrder(p, base) is never p1 for any prime p Last fiddled with by robert44444uk on 20070825 at 04:25 
20130820, 12:58  #8 
Feb 2003
1,907 Posts 
k=13833343704 is GCC(11,)11 for n=0...10.

20130820, 17:58  #9 
Feb 2003
1,907 Posts 
And a few GCC(3,)10:
For n=0 to 9: k= 1030544270 16540413680 62072286920 62683142060 98303255750 For n=1 to 10: k= 10692363780 10749790380 25120807810 45213014140 
20130822, 17:11  #10 
Jun 2003
Oxford, UK
11×173 Posts 
I thought I would get in with one or two more b=4
GCC(4,)12 for n =0 to 11 Code:
3123824802 10808693852 38264032488 Code:
2702173463 9566008122 for n=0 to 10 Code:
161205842 1661154150 4492296738 8870650620 12495299208 43234775408 44088473310 44222466372 Code:
1847901660 2217662655 8288593367 11055616593 34499196413 46649435007 
20130822, 17:49  #11 
Feb 2003
1,907 Posts 
And a few GCC(6,)10:
For n=0 to 9: k=5877226322 For n=1 to 10: k=566408953, 2049564484 Last fiddled with by Thomas11 on 20130822 at 17:54 
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