- Could somebody please point me in the direction where

I can find out the current records for Cunningham

Chains length>5.

I have looked at

http://ksc9.th.com/warut/cunningham.html

but it doesn't appear to have been updated for quite

some time.

Any ideas?

Gary

__________________________________________________

Do You Yahoo!?

Everything you'll ever need on one web page

from News and Sport to Email and Music Charts

http://uk.my.yahoo.com - Gary,

Dirk Augustin maintains a file at

http://groups.yahoo.com/group/primenumbers/files/Prime%20Tables/Cunningham_Cha

in_records.txt

Regards,

Paul.

__________________________________________________

Virus checked by MessageLabs Virus Control Centre. - This may be a silly question, but:

Why not have chains that are closer to half, rather than almost double?

Would not the chains then be longer than Cunningham's?

Example: 2,3,5,7,11,13,19 immediately produces a 7 chain, on a closer to half basis, and I feel heuristically longer ones should be readily available. Also, are mixed chains of the 1st and 2nd Cunningham kind allowed? Or, have I missed the reason for the purpose of the originals?

Bob

[Non-text portions of this message have been removed] - Bob Gilson wrote:
> Why not have chains that are closer to half, rather than

I assume your rule is the next number is the closest odd

> almost double?

> Would not the chains then be longer than Cunningham's?

> Example: 2,3,5,7,11,13,19 immediately produces a 7 chain,

> on a closer to half basis, and I feel heuristically longer

> ones should be readily available. Also, are mixed chains

> of the 1st and 2nd Cunningham kind allowed? Or, have I

> missed the reason for the purpose of the originals?

> Sorry I meant 2.3.5.7.11.17 a 6 chain in my question

number to 1.5p. I don't think this makes longer chains easier,

except maybe a small advantage from numbers growing slower.

Cunningham chains have been generalized in different ways

but they don't appear to have attracted much attention.

Here are two ways:

http://www.primenumbers.net/Henri/us/CunnGenus.htm

http://www.primepuzzles.net/puzzles/puzz_060.htm

Mixed normal chains of the 1st and 2nd kind with start p > 3

are impossible due to divisibility by 3.

If p and 2p+1 are prime then p is 2 modulo 3, and 3 divides

2*(2p+1)-1.

Similarly, if p and 2p-1 are prime then 3 divides 2*(2p-1)+1.

But the generalized Cunningham chains in line 1 of

http://www.primenumbers.net/Henri/us/CunnGenus.htm

can be altered to allow mixed signs, resulting in "prime trees":

http://unbecominglevity.blogharbor.com/blog/_archives/2004/3/17/27759.html

Mixing + and - in this way gives more possibilities and

longer "chains" can be found.

2p+/-308843535 starting at p=177857809 gives a prime tree

of depth 26:

http://unbecominglevity.blogharbor.com/blog/_archives/2006/5/12/1952529.html

The corresponding "chain" of length 26 where d=308843535:

Start number: 177857809

* 2 - d = 46872083

* 2 + d = 402587701

* 2 + d = 1114018937

* 2 + d = 2536881409

* 2 - d = 4764919283

* 2 - d = 9220995031

* 2 + d = 18750833597

* 2 - d = 37192823659

* 2 - d = 74076803783

* 2 - d = 147844764031

* 2 - d = 295380684527

* 2 + d = 591070212589

* 2 - d = 1181831581643

* 2 + d = 2363972006821

* 2 + d = 4728252857177

* 2 - d = 9456196870819

* 2 + d = 18912702585173

* 2 + d = 37825714013881

* 2 + d = 75651736871297

* 2 + d = 151303782586129

* 2 + d = 302607874015793

* 2 - d = 605215439188051

* 2 - d = 1210430569532567

* 2 - d = 2420860830221599

* 2 + d = 4841721969286733

--

Jens Kruse Andersen