Problems & Puzzles:
Puzzles
Puzzle 119. SophieGermain & Twin, chains
In this puzzle we ask for chains of primes
P_{1}, P_{2}, P_{3}, ... 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
Length 
First prime of the early
chain 
By, Year 
2 
5 
H.Lifchitz, 98 
3 
211049 (1005*7# 1) 
H.Lifchitz, 98 
4 
253679 (151*7#*2^31) 
H.Lifchitz, 98 
5 
1394847*13# +/1 
Jack Brennen, 99 
6 
1228253271*13#*2 +/1 
Jack Brennen, 99 
7 
11228462199623*13# +/1 
Paul Jobling, 99 
8 
? 
Paul Jobling &
Phil Carmody, 2002 (see below) 
***
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/frus/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, 2q1, ...} 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
primechains?
***
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 pseudoprimality 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:
Hola Carlos, A while ago myself and Phil Carmody
found a chain of length 8: 21033215071024191*13# *2^k + 1, k=0 to 7. We
are not sure if that is the smallest, however. Regards, Paul
So, the game  the initial one  is
over...
***
