- Could somebody please point me in the direction where
I can find out the current records for Cunningham
I have looked at
but it doesn't appear to have been updated for quite
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- 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?
[Non-text portions of this message have been removed]
- Bob Gilson wrote:
> Why not have chains that are closer to half, rather thanI 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 188.8.131.52.11.17 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:
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
Similarly, if p and 2p-1 are prime then 3 divides 2*(2p-1)+1.
But the generalized Cunningham chains in line 1 of
can be altered to allow mixed signs, resulting in "prime trees":
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:
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