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Old 2007-08-14, 10:20   #1
robert44444uk
 
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Default 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
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Old 2007-08-15, 07:14   #2
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Default 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
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Old 2007-08-16, 08:48   #3
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Default GCC(4-)13!!!!!

Found my first chain of 13,

6703351518*4^n-1, n from 1 to 13 all prime!!
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Old 2007-08-22, 06:46   #4
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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
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Old 2007-08-22, 14:04   #5
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Default

Quote:
Originally Posted by robert44444uk View Post
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?
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Old 2007-08-24, 14:32   #6
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Quote:
Originally Posted by wblipp View Post
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.
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Old 2007-08-25, 04:25   #7
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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
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Old 2013-08-20, 12:58   #8
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k=13833343704 is GCC(11,-)11 for n=0...10.
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Old 2013-08-20, 17:58   #9
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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
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Old 2013-08-22, 17:11   #10
robert44444uk
 
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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
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Old 2013-08-22, 17:49   #11
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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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