## Consecutive occurrences of decadal prime triplets

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• A decadal prime triplet is a set matching one of the four types {10K+1, 10K+3, 10K+7}, {10K+1, 10K+3, 10K+9}, {10K+1, 10K+7, 10K+9}, {10K+3 10K+7, 10K+9},
Message 1 of 7 , Dec 31, 2009
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A decadal prime triplet is a set matching one of the four types {10K+1,
10K+3, 10K+7}, {10K+1, 10K+3, 10K+9}, {10K+1, 10K+7, 10K+9}, {10K+3
10K+7, 10K+9}, all members of course being prime. Consecutive
occurrences have a gap of 30 (i.e. the K value is incremented by 3), and
in this context quadruplets are permitted as well.

There is an instance of 4 consecutive triplets at

5413 5417 5419; 5441 5443 5449; 5471 5477 5479; 5501 5503 5507.

By choosing suitable values modulo some small prime divisors, it appears
to be possible to have up to 11 consecutive occurrences of such
triplets. There is not even another case of 4 up to about 25000, from a
quick check. Before I invest resources trying to do very long searches
for any other runs of 4 or better, I am interested to know if anyone has
covered this ground already. Any information is welcome.

[Non-text portions of this message have been removed]
• ... I have not heard of previous searches for this. I confirm the largest admissible patterns have 11 consecutive triplets. The number of admissible patterns
Message 2 of 7 , Dec 31, 2009
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rupert.wood wrote:
> A decadal prime triplet is a set matching one of the four types {10K+1,
> 10K+3, 10K+7}, {10K+1, 10K+3, 10K+9}, {10K+1, 10K+7, 10K+9}, {10K+3
> 10K+7, 10K+9}, all members of course being prime. Consecutive
> occurrences have a gap of 30 (i.e. the K value is incremented by 3), and
> in this context quadruplets are permitted as well.
>
> There is an instance of 4 consecutive triplets at
>
> 5413 5417 5419; 5441 5443 5449; 5471 5477 5479; 5501 5503 5507.
>
> By choosing suitable values modulo some small prime divisors, it appears
> to be possible to have up to 11 consecutive occurrences of such
> triplets. There is not even another case of 4 up to about 25000, from a
> quick check. Before I invest resources trying to do very long searches
> for any other runs of 4 or better, I am interested to know if anyone has
> covered this ground already. Any information is welcome.

I have not heard of previous searches for this.
I confirm the largest admissible patterns have 11 consecutive triplets.
The number of admissible patterns with 1 to 11 triplets:
4, 16, 44, 85, 194, 229, 228, 210, 100, 26, 4.

One of the 4 admissible patterns with 11 triplets:
{3, 7, 9, 31, 33, 37, 61, 63, 69, 91, 93, 97, 121, 123, 129,
151, 153, 159, 181, 187, 189, 213, 217, 219, 241, 247, 249,
271, 273, 279, 301, 303, 307}

I am not searching the next cases with 4 triplets so you can do that if
you want. I guess you don't have suitable tools to search for 5 triplets.

The smallest instance of any of the 194 patterns with 5 triplets:
9100524636850 + n, for n =
1, 7, 9, 31, 33, 37, 61, 63, 69, 91, 93, 99, 121, 127, 129.
It is also prime for n=21 and no others in the interval.

Two larger instances of other patterns with 5 triplets:

283955584598830 + n, for n =
1, 7, 9, 33, 37, 39, 61, 63, 69, 91, 93, 97, 121, 123, 127.
Also prime for n=19.

488187768695650 + n, for n =
1, 7, 9, 31, 37, 39, 63, 67, 69, 93, 97, 99, 121, 127, 129.
Also prime for n=27 and n=57.

I don't know whether there are other patterns with smaller instances.

--
Jens Kruse Andersen
• ... My search says these are the first three instances of 5 triplets. I don t have resources to search for 6 triplets. -- Jens Kruse Andersen
Message 3 of 7 , Dec 31, 2009
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I wrote:
> 9100524636850 + n, for n =
> 1, 7, 9, 31, 33, 37, 61, 63, 69, 91, 93, 99, 121, 127, 129.
>
> 283955584598830 + n, for n =
> 1, 7, 9, 33, 37, 39, 61, 63, 69, 91, 93, 97, 121, 123, 127.
>
> 488187768695650 + n, for n =
> 1, 7, 9, 31, 37, 39, 63, 67, 69, 93, 97, 99, 121, 127, 129.

My search says these are the first three instances of 5 triplets.
I don't have resources to search for 6 triplets.

--
Jens Kruse Andersen
• Does it not matter that primes exist between them? For instance 9100524636850+n is also prime for n=21 [Non-text portions of this message have been removed]
Message 4 of 7 , Dec 31, 2009
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Does it not matter that primes exist between them? For instance 9100524636850+n is also prime for n=21

[Non-text portions of this message have been removed]
• ... {10K+1, ... and ... appears ... from a ... searches ... has ... triplets. ... if ... triplets. ... Thanks very much for the information in that and your
Message 5 of 7 , Dec 31, 2009
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--- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
<jens.k.a@...> wrote:
>
> rupert.wood wrote:
> > A decadal prime triplet is a set matching one of the four types
{10K+1,
> > 10K+3, 10K+7}, {10K+1, 10K+3, 10K+9}, {10K+1, 10K+7, 10K+9}, {10K+3
> > 10K+7, 10K+9}, all members of course being prime. Consecutive
> > occurrences have a gap of 30 (i.e. the K value is incremented by 3),
and
> > in this context quadruplets are permitted as well.
> >
> > There is an instance of 4 consecutive triplets at
> >
> > 5413 5417 5419; 5441 5443 5449; 5471 5477 5479; 5501 5503 5507.
> >
> > By choosing suitable values modulo some small prime divisors, it
appears
> > to be possible to have up to 11 consecutive occurrences of such
> > triplets. There is not even another case of 4 up to about 25000,
from a
> > quick check. Before I invest resources trying to do very long
searches
> > for any other runs of 4 or better, I am interested to know if anyone
has
> > covered this ground already. Any information is welcome.
>
> I have not heard of previous searches for this.
> I confirm the largest admissible patterns have 11 consecutive
triplets.
> The number of admissible patterns with 1 to 11 triplets:
> 4, 16, 44, 85, 194, 229, 228, 210, 100, 26, 4.
>
> One of the 4 admissible patterns with 11 triplets:
> {3, 7, 9, 31, 33, 37, 61, 63, 69, 91, 93, 97, 121, 123, 129,
> 151, 153, 159, 181, 187, 189, 213, 217, 219, 241, 247, 249,
> 271, 273, 279, 301, 303, 307}
>
> I am not searching the next cases with 4 triplets so you can do that
if
> you want. I guess you don't have suitable tools to search for 5
triplets.
>
> The smallest instance of any of the 194 patterns with 5 triplets:
> 9100524636850 + n, for n =
> 1, 7, 9, 31, 33, 37, 61, 63, 69, 91, 93, 99, 121, 127, 129.
> It is also prime for n=21 and no others in the interval.
>
> Two larger instances of other patterns with 5 triplets:
>
> 283955584598830 + n, for n =
> 1, 7, 9, 33, 37, 39, 61, 63, 69, 91, 93, 97, 121, 123, 127.
> Also prime for n=19.
>
> 488187768695650 + n, for n =
> 1, 7, 9, 31, 37, 39, 63, 67, 69, 93, 97, 99, 121, 127, 129.
> Also prime for n=27 and n=57.
>
> I don't know whether there are other patterns with smaller instances.
>
> --
> Jens Kruse Andersen
>

Thanks very much for the information in that and your other post Jens. I
don't have the resources to search for 5 triplets. I hope to be able to
do so for 4 triplet case once I finally get my issues with gp sorted
out.
• ... Thanks for that! Just to avoid unnecessary effort and extra computer time, are there any coding shortcuts for this kind of searching? Even in the 4-triplet
Message 6 of 7 , Jan 1, 2010
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--- In primenumbers@yahoogroups.com, "Jens Kruse Andersen" <jens.k.a@...> wrote:
>
> I wrote:
> > 9100524636850 + n, for n =
> > 1, 7, 9, 31, 33, 37, 61, 63, 69, 91, 93, 99, 121, 127, 129.
> >
> > 283955584598830 + n, for n =
> > 1, 7, 9, 33, 37, 39, 61, 63, 69, 91, 93, 97, 121, 123, 127.
> >
> > 488187768695650 + n, for n =
> > 1, 7, 9, 31, 37, 39, 63, 67, 69, 93, 97, 99, 121, 127, 129.
>
> My search says these are the first three instances of 5 triplets.
> I don't have resources to search for 6 triplets.
>
> --
> Jens Kruse Andersen
>

Thanks for that!

Just to avoid unnecessary effort and extra computer time, are there any coding shortcuts for this kind of searching? Even in the 4-triplet case there would be quite a bit of tedious checking to do in each iteration (just asking in case someone has developed some generic sort of prime pattern searching routine).
• ... I used my own unpublished prime pattern finder. It is modified for each search and not suited for sharing. It could probably easily find thousands of
Message 7 of 7 , Jan 2, 2010
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rupert.wood wrote:
> Just to avoid unnecessary effort and extra computer time,
> are there any coding shortcuts for this kind of searching?
> Even in the 4-triplet case there would be quite a bit of
> tedious checking to do in each iteration (just asking in
> case someone has developed some generic sort of prime
> pattern searching routine).

I used my own unpublished prime pattern finder. It is modified for
each search and not suited for sharing. It could probably easily
find thousands of 4-triplet cases if it was modified for the purpose.

There are many possible shortcuts evolving around avoiding or quickly
eliminating cases where at least one number has a small prime factor.
I searched each of the 194 admissible 5-triplet patterns one at a time,
so in each case there were 15 numbers that had to be prime. Searching
some patterns with few differences at the same time might be more
efficient but I didn't have suitable code for that.

A shortcut you may already use is to only make prp (probable prime)
tests at first, and only make primality proofs later when there is a
complete prp solution.

Using fast tools like C instead of PARI/GP can also speed up many things.

Robin Garcia wrote:
> Does it not matter that primes exist between them?
> For instance 9100524636850+n is also prime for n=21

I listed this and the primes for the other cases in my first post