## Re: [PrimeNumbers] Incidence of cunningham chains and twins

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• Do the CCs occur in roughly equal numbers or is a particular sort of CC more common?!? Gary Robert wrote: In my search for triple
Message 1 of 4 , Jul 4, 2005
Do the CCs occur in roughly equal numbers or is a particular sort of CC more common?!?
Gary

Robert <rw.smith@...> wrote:
In my search for "triple double" values, I am looking for a k, such
that the two power series k.2^n+&-1 provide more than 10 Cunningham
Chains (1st kind), 10 Cunningham Chains (2nd kind) and 10 twins.
during the search, I have noticed that the number of k's producing
more than 10 Cunningham Chains length 2 (1st or 2nd kind) tends to
be consistently higher than the number of k providing 10 twins, and
the maxima of CCs found also looks to be higher than the maxima for
twins.

In this arrangement I count a Cunningham Chain of length 3 as two
CC's length 2, a CC4 as 3 CC2's etc.

Is there a fundamental reason for this? I would have thought that
for a given prime of the form k.2^n+1, that there were slightly
higher chances of k.2^n-1 being prime than k.2^(n+1)+1, given that
the twin partner is smaller than the CC partner. It would follow, to
my untutored, peanut sized brain that there would be more twins as a
result than CC's for a given power series.

The k's I am checking are chosen as follows:- they are multiples of
29#/17, and the n values 1 to 10 do not have any factors smaller
than 256, + or -.

I only take forward for checking the values of k which provide at
least 7 "triple double" points in the first 10 n, and 13 points by
n=100.

I will be happy to provide anyone who wants it with further
statistics to support the observations, made over the values from
k=1 up to k= 30trillion*29#/17

Regards

Robert Smith

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• From: Robert ... My immediate gut feel reaction tells me this would be expected. I suspect that my gut feel is wrong though! As
Message 2 of 4 , Jul 5, 2005
From: "Robert" <rw.smith@...>
> In my search for "triple double" values, I am looking for a k, such
> that the two power series k.2^n+&-1 provide more than 10 Cunningham
> Chains (1st kind), 10 Cunningham Chains (2nd kind) and 10 twins.
> during the search, I have noticed that the number of k's producing
> more than 10 Cunningham Chains length 2 (1st or 2nd kind) tends to
> be consistently higher than the number of k providing 10 twins, and
> the maxima of CCs found also looks to be higher than the maxima for
> twins.

My immediate gut feel reaction tells me this would be expected.
I suspect that my gut feel is wrong though!
As always, everything should be predicted by sieving with small primes so
look at residues of k*2^n+/-1 mod p for small p, fixed k, variable n.

Phil

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• ... CC more common?!? ... Gary: No simple answer to that. I would have to rerun the candidates to find out how many CC3, CC4 etc. My program only counts
Message 3 of 4 , Jul 5, 2005
--- In primenumbers@yahoogroups.com, Gary Chaffey <garychaffey2@y...>
wrote:
> Do the CCs occur in roughly equal numbers or is a particular sort of
CC more common?!?
> Gary

Gary: No simple answer to that. I would have to rerun the candidates
to find out how many CC3, CC4 etc. My program only counts points.

The statistics for the range of k 1 to 1531174737133 multiplied by
29#/34 (approx 1/20th of the overall numbers checked):

25250 values of k have 7 or more triple double points at n=10
of these:
2814 had 13 or more points at n=100 and of these:
81 had 20 or more points at n=500, the best of which were 4 23's and 5
22's

49 of those taken to 500 (2814 candidates)had 10 or more CC's of the
1st kind (including a 13 and 3 12's)
41 of those taken to 500 had 10 or more CCs of the 2nd kind (including
7 12's)
4 of those taken to 500 had 10 or more twins (1 with 11)
1 candidate fell into two categories

As I might have expected, twins are slightly more prevalent - if I
average the scores of the 2814 candidates with 13 or more points, they
have 5.09 CC1s, 5.09 CC2s and 5.25 twins.

So it appears that the distribution curves for CC's amd twins are not
the same, which defies simple analysis.

Regards

Robert Smith
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