- 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] - rupert.wood wrote:
> Just to avoid unnecessary effort and extra computer time,

I used my own unpublished prime pattern finder. It is modified for

> 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).

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?

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

> For instance 9100524636850+n is also prime for n=21

http://tech.groups.yahoo.com/group/primenumbers/message/21242

The original post said prime quadruplets are permitted and also listed:

> There is an instance of 4 consecutive triplets at

5431, 5437 and 5483 are also prime.

>

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

--

Jens Kruse Andersen