Problems & Puzzles:
Puzzles
Puzzle 122. Consecutive Twin primes
A natural and direct consequence of the Puzzle
121 is to ask by:
1) Chains of K Consecutive Twin
primes in A.P.
2) Chains of K Consecutive
Twin primes, without any further condition about the separation
between adjacent pairs.
K pairs of twins (p_{1}, p_{1}+2),
(p_{2}, p_{2}+2), ....(p_{k}, p_{k}+2) are "consecutive" if there are
are not prime numbers between p_{1} & p_{k}+2 other
than the corresponding to the twin primes inside the interval.
Examples:
a) The trio of twin primes {(4217,4219),
(4229,4231), (4241, 4243)} are consecutive because there are only
composite numbers between 4217& 4243 except those belonging to the
shown twins. Additionally this trio is in A.P., step 12.
b) The quartet if twin primes {(9419,
9421), (9431,9433), (9437,9439), (9461,9463)} are consecutive because
there are only composite numbers between 9419 & 9463, except... But this quartet is not in A.P.
The
following two tables summarizes what I have found:
Table 1
Consecutive twin primes in A.P.
Red color the
first prime of the least sequence for each K & Step values. 
K 
Step 
First Prime 
Author 
3 
6 
5 

3 
12 
4217

C.R.,
12/2000 
3 
30 
208931 
C.R.,
12/2000 
4 
30 
263872067

C.R.,
12/2000 
5 
30 
127397154761 
Phil Carmody, 7/1/01 
6 



7 



Table
2
Consecutive twin primes
(There is not necessarily a common difference) 
K

Shown
the lower prime of each pair for the least sequence in each K

Author 
3

5,
11, 17


4

9419,
9431, 9437, 9461

C.R.,
12/2000 
5

909287,
909299, 909317, 909329, 909341

C.R.,
12/2000 
6

325267931,
325267937, 325267949, 325267961, 325267979, 325267991

C.R.,
12/2000 
7

678771479,
678771491, 678771551, 678771557, 678771617, 678771647, 678771659

C.R.,
12/2000 
8

1107819732821
, 1107819732911, 1107819732917,
1107819732947, 1107819732959,
1107819732977, 1107819733037,
1107819733061 
Phil
Carmody, 8/1/01 
9 


10 


Question: Would you like
to extend the Tables 1 & 2?
Jud McCranie wrote (6/1/01):
"part 1  no solutions < 55,000,000,000 for larger k
part 2  no solutions <
100,000,000,000 for larger k"
***
The first new & positive result became from Phil
Carmody who found (7/1/01) the earlier chain of 5 consecutive twins in A.P.
He also wrote "I've officially made this puzzle
one of the test programs for the prime generator to be used in the forthcoming
versions of PFGW (what was PrimeForm)"
***
Phil Carmody also got (8/1/01) the first example of 8
twins just consecutive (See table above). He added this time
"The total sieve has been exhaustive, from zero with no gaps, so
nothing needs to be checked below my limit. I reached 1,138,166,333,443."
***
Denis DeVries has the record (28/3/2002)!!!:
9
consecutive twins not in A.P.:
170669145704411 170669145704413
170669145704501 170669145704503 170669145704507
170669145704509 170669145704591 170669145704593
170669145704639 170669145704641 170669145704669
170669145704671 170669145704747 170669145704749
170669145704807 170669145704809 170669145704819
170669145704821
How he found them?
"I started looking for twin primes several years
ago using a sieve program developed for 64k machines in 1985. Pascal
Sebah (pascal_sebah@dsfr.com)
announced some results on Brun's constant at numbrthry@Listserve.nodak.edu
& sent me his code which was much faster than mine. I don't know
how it works as I only have the compiled version which he modified for my
search"
***
Nine months later Denis
(24/12/2002) wrote again:
While searching for the 1st set of 10 consecutive
twins, I found the 2nd set of 9. It's at 59765 55030 30737 thru 31049.
I suspect the 1st occurrence of a ten set is in the
range 5E15 to 6E15. I don't expect to find it.
***
Gabor Levai
found (July 2004) two more examples of 9 consecutive twins:
I found 9 consecutive twins in the intervals
1) [4518517172328671,4518517172329009]
2) [1980326398382819,1980326398383373]
...
Finally, on September 13, 2004 Gabor
wrote:
I found the first 10 consecutive twins
in [3324648277099157,3324648277099453], 16 digits:
3324648277099157,3324648277099159
3324648277099211,3324648277099213 3324648277099229,3324648277099231
3324648277099241,3324648277099243 3324648277099307,3324648277099309
3324648277099319,3324648277099321 3324648277099337,3324648277099339
3324648277099397,3324648277099399 3324648277099421,3324648277099423
3324648277099451,3324648277099453
Because of parallel computing on
different intervals currently I'm not sure
that this is the earliest.
The full interval is [ 1, 2^52 ], 1018 computer work
with very different speed.
There are 20 uncovered range before the 10 twins. Currently ~78% is scanned.
Because of a wrong parameter one computer not stopped at 2^52, the result
are the 9 twins in [4518517172328671,4518517172329009].
***
On July 9, 2006, Gabor wrote:
I found the 1st set of 10 consecutive twins greater
than 2^63 in the interval
[9224121946845515441,9224121946845515863], 19 digits:
..441, ..443; ..537, ..539; ..627, ..629; ..639,
..641; ..657, ..659;
..711, ..713; ..777, ..779; ..819, ..821; ..849, ..851; ..861, ..863;
***
Later on Feb. 07, Gabor wrote again:
Two new interval with 10 consecutive twin primes:
[31910610414019031,31910610414019459], 17 digits,
[9226245365154613667,9226245365154614101], 19 digits.
***
On October 2011, Gabor wrote again:
I found the first 11 consecutive twin primes in the
interval: [789795449254776509,
789795449254776871].
***
