Generalised Cunningham Chains

robert44444uk · 2007-08-14, 10:20

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

robert44444uk · 2007-08-15, 07:14

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

robert44444uk · 2007-08-16, 08:48

Found my first chain of 13,

6703351518*4^n-1, n from 1 to 13 all prime!!

robert44444uk · 2007-08-22, 06:46

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

wblipp · 2007-08-22, 14:04

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?

robert44444uk · 2007-08-24, 14:32

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.

robert44444uk · 2007-08-25, 04:25

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

Thomas11 · 2013-08-20, 12:58

k=13833343704 is GCC(11,-)11 for n=0...10.

Thomas11 · 2013-08-20, 17:58

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

robert44444uk · 2013-08-22, 17:11

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

Thomas11 · 2013-08-22, 17:49

And a few GCC(6,-)10:

For n=0 to 9: k=5877226322
For n=1 to 10: k=566408953, 2049564484

2007-08-14, 10:20	#1
robert44444uk Jun 2003 Suva, Fiji 2·1,021 Posts	Generalised Cunningham Chains A Cunningham Chain length 2 is k2^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: 103477472709803^n+1, n from 1 to 10 all prime = GCC(3)10 5503265885^n+1, n from 1 to 10 = GCC(5)10 6789799044607^n+1, n from 1 to 9 = GCC(7)9 94315197611^n+1, n from 1 to 9 = GCC(11)9 292402788023^n+1, n from 1 to 9 = GCC(23)9 9163669086023^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 Suva, Fiji 2×1,021 Posts	GCC(4)11! 954726224^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 Suva, Fiji 2×1,021 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 Suva, Fiji 2×1,021 Posts	Using the notation GCC(base, + or -)"length of Generalised Cunningham Chain" for the form kbase^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*

2007-08-25, 04:25	#7
robert44444uk Jun 2003 Suva, Fiji 2×1,021 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 2⁷·3·5 Posts	k=13833343704 is GCC(11,-)11 for n=0...10.

2013-08-20, 17:58	#9
Thomas11 Feb 2003 2⁷·3·5 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 Suva, Fiji 11111111010₂ 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 2⁷×3×5 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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