Problems & Puzzles: Puzzles

Puzzle 572.- (P+Q)@3<>0

JM Bergot sent the following puzzle:

Look for consecutive primes such that added every two contiguous the sum is not divided by 3

I [CR] got 21 of such sums with the following 22 consecutive primes & the minimal for this run:

766319189 766319201 766319231 766319237 766319249 766319261 766319273 766319291 766319339 766319357 766319363 766319369 766319423 766319441 766319453 766319483 766319507 766319549 766319573 766319579 766319621 766319627

Q. Can you find a largest run of these type of consecutive primes (if the minimal for the run, the better)

Contributions came from J. K. Andersen, Seiji Tomita, Giovanni Resta, Farideh Firoozbakht.

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J. K. Andersen wrote:

This is the same as puzzle 448. The longest known run is still 31 primes starting at 1552841185921. (See Giovanni Resta contribution below)

My page has moved tohttp://users.cybercity.dk/~dsl522332/math/congruent-primes.htm#mod6

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Seiji Tomita wrote:

I found a same type of 24 cosecutive primes such as (P+Q)@4<>0.
All cosecutive primes have the form of 4m+1.

1113443017, 1113443029, 1113443053, 1113443069, 1113443077
1113443137, 1113443141, 1113443153, 1113443173, 1113443189
1113443209, 1113443249, 1113443269, 1113443293, 1113443329
1113443341, 1113443369, 1113443381, 1113443389, 1113443473
1113443521, 1113443533, 1113443549, 1113443561

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Giovanni Resta wrote:

Puzzle 572 is actually equal to Puzzle 448. So I tried to extend Puzzle 448.

I found a run of 32 primes, from 24,738,041,398,529 to 24,738,041,399,471
which are equal to 2 mod 3.

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Farideh Firoozbakht wrote:

According to the latest known term of the sequence A055626 we find the following chain of 25 consecutive primes of the form 3k+2, starting with A055626(25) = 21549657539.

21549657539,21549657551,21549657581,21549657587,21549657611,
21549657617,21549657641,21549657653,21549657659,21549657737,
21549657749,21549657791,21549657809,21549657839,21549657863,
21549657893,21549657911,21549657917,21549657947,21549657953,
21549657959,21549657989,21549658007,21549658037,21549658079

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On Jan 22, 2011, Giovanni Resta wrote:

I found another longer run (and now I stopped my search):

the 35 primes from 41173225034771 to 41173225035809
are equal to 2 mod 3.

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