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 2007-08-14, 10:20 #1 robert44444uk     Jun 2003 Oxford, UK 1,907 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+(y-1) 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 2007-08-14 at 10:23
 2007-08-15, 07:14 #2 robert44444uk     Jun 2003 Oxford, UK 1,907 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 2007-08-15 at 07:17
 2007-08-16, 08:48 #3 robert44444uk     Jun 2003 Oxford, UK 1,907 Posts GCC(4-)13!!!!! Found my first chain of 13, 6703351518*4^n-1, n from 1 to 13 all prime!!
 2007-08-22, 06:46 #4 robert44444uk     Jun 2003 Oxford, UK 1,907 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 2007-08-22 at 07:05
2007-08-22, 14:04   #5
wblipp

"William"
May 2003
New Haven

23·5·59 Posts

Quote:
 Originally Posted by robert44444uk 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
I don't understand the terminology.

1081477811*9n+1 is always even. What is the correct expression for the primes?

2007-08-24, 14:32   #6
robert44444uk

Jun 2003
Oxford, UK

1,907 Posts

Quote:
 Originally Posted by wblipp I don't understand the terminology. 1081477811*9n+1 is always even. What is the correct expression for the primes?
Aiaia, sorry, all k values quoted for the GCC(9,+)9 are to be multiplied by 2. Didn't spot that in my program output.

 2007-08-25, 04:25 #7 robert44444uk     Jun 2003 Oxford, UK 35638 Posts k=84378963 is GCC(16,+)11 Bases that are squares are particularly rich as ModuloOrder(p, base) is never p-1 for any prime p Last fiddled with by robert44444uk on 2007-08-25 at 04:25
 2013-08-20, 12:58 #8 Thomas11     Feb 2003 35638 Posts k=13833343704 is GCC(11,-)11 for n=0...10.
 2013-08-20, 17:58 #9 Thomas11     Feb 2003 190710 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
 2013-08-22, 17:11 #10 robert44444uk     Jun 2003 Oxford, UK 1,907 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 for n=1 to 12 Code: 2702173463 9566008122 and GCC(4,-)11 for n=0 to 10 Code: 161205842 1661154150 4492296738 8870650620 12495299208 43234775408 44088473310 44222466372 for n=1 to 11 Code: 1847901660 2217662655 8288593367 11055616593 34499196413 46649435007
 2013-08-22, 17:49 #11 Thomas11     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 2013-08-22 at 17:54

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