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 (p1, p1+2), (p2, p2+2), ....(pk, pk+2) are "consecutive" if there are are not prime numbers between p1 & pk+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"

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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)"

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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."

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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@ds-fr.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"

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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.

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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 ], 10-18 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].

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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;

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Later on Feb. 07, Gabor wrote again:

Two new interval with 10 consecutive twin primes:

[31910610414019031,31910610414019459], 17 digits,
[9226245365154613667,9226245365154614101], 19 digits.

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On October 2011, Gabor wrote again:

I found the first 11 consecutive twin primes in the interval: [789795449254776509, 789795449254776871].

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