Problems & Puzzles: Puzzles

Puzzle 119. Sophie-Germain & Twin, chains

In this puzzle we ask for chains of primes P₁, P₂, P₃, ... such that:

a) P_i+1=2.P_i+1 &
b) Each Pi is the small prime of a Twin prime pair.

Here is an easy example and the early of length 4:  {253679, 507359, 1014719, 2029439}

Question: Find the early chain of length 5, 6, 7 & 8 
_______
Note: This puzzle came to me after Mr. John Everett sent to me a magic prime square 4x4 with all the primes having this property (SG & Twin). Thanks John for the idea.

Solution

Status of the search

***

Knowing that the popular sieving code NewPGen by Paul Jobling deals with some chains of primes like the ones asked in this puzzle, I wrote this morning to Paul the following:

"Is your popular sieving code useful for this puzzle...According to you, how should be properly called these kind of chains? Do you know if this type larger sequences have been published somewhere?"

This is his immediate answer:

"Yes, NewPGen can indeed find these - they are called "BiTwin chains". Currently the longer known is of length 7, found by myself some time ago, so perhaps your puzzlers might be able to do better. Here is a page full of records:

http://ourworld.compuserve.com/homepages/hlifchitz/Henri/fr-us/BiTwinRec.htm

(Note that Henri counts the links, which is 1 less than the number of members of the chain)

This also lists the earliest for the lengths up to 7. Note that I wrote some special software to find these long chains, NewPGen is really only good for finding large short chains. I reckon that the earliest of length 8 could be found in about 1 or 2 months with my software - it took it 24 hours to find the smallest of length 7.

Note that these can be viewed as two related Cunningham Chains of the first and second kind: {p, 2p+1, ...) and {q, 2q-1, ...} where q=p+2."

So, shortly & suddenly the puzzle was reduced to only one question: what is the early chain of length 8?

Now we know: a)what is the used name of these chains b)who produced the early cases for the first 7 chains c)who (Henri Lifchitz) keeps the status of these chains and c)that one tool to find these chains is the Jobling's code. Is there any other public/free tool available to search for this kind of prime-chains?

***

Paul Jobling has sent (17/12/2000) a specific code executable to do this search. I will sent it by email on request.

This is the Paul's email:

You ask whether any software is available to perform this search. I have attached the program that I wrote to search for long BiTwin chains and also long Cunningham Chains. I would imagine that the other people who have written this sort of software would be Jack Brennen and Tony Forbes. I have no problems with this software being redistributed.

 

It searches through numbers of the form i*K*2^n+1, where n goes from 0 to the length of the chain -1 that you are after, and K is an expression such as 13#.... On this PC (a 500 MHz Pentium III) it can search for the smallest of length 8 at a rate of 155 million i's per second. This includes sieving against 1000 small primes, and then performing Fermat pseudo-primality tests on those i's that get through the sieve.

If somebody does use my software to look for a record, they should start from i=11228462199623 to save about a days worth of processing, as I have checked up to there

***

The interesting day 11/11/2002, Paul Jobling wrote:

So, the game - the initial one - is over...

***