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Re: [PrimeNumbers] Interesting prime sequence

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  • Phil Carmody
    ... So basically one sieves out -1...-L (mod p) for each p not a factor of any 1..L. And one sieves _in_ 0 mod p otherwise? Sounds, erm, doable... Phil
    Message 1 of 6 , Mar 27, 2002
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      --- jbrennen <jack@...> wrote:
      > As an interesting little project, I've been looking for
      > sequences of consecutive integers where N+1 is prime,
      > N+2 is 2 times a prime, N+3 is 3 times a prime, etc.
      >
      > Another way to look at it is to find an integer N that
      > maximizes the number of primes in the sequence
      > N+1, N/2+1, N/3+1, N/4+1, ...
      >
      > I've been searching for the smallest such N for each
      > successive length:
      >
      > Length N
      > ------ -
      > 1 1
      > 2 2
      > 3 12
      > 4 12720
      > 5 19440
      > 6 5516280
      > 7 5516280
      > 8 7321991040
      > 9 363500177040
      > 10 2394196081200
      > 11 3163427380990800
      >
      >
      > I'm currently sieving through about 7 million candidates/second
      > while looking for the smallest such sequence of length 12.
      > I'm sure that sieve maestro Phil could beat that quite easily,
      > with his Alpha sieve programming skills, but it sure beats
      > doing trial division :-)

      So basically one sieves out -1...-L (mod p) for each p not a factor
      of any 1..L. And one sieves _in_ 0 mod p otherwise?

      Sounds, erm, doable...

      Phil

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    • Phil Carmody
      ... Alright, just for you I ran on an x86 machine...
      Message 2 of 6 , Mar 28, 2002
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        --- jbrennen <jack@...> wrote:
        > As an interesting little project, I've been looking for
        > sequences of consecutive integers where N+1 is prime,
        > N+2 is 2 times a prime, N+3 is 3 times a prime, etc.
        >
        > Another way to look at it is to find an integer N that
        > maximizes the number of primes in the sequence
        > N+1, N/2+1, N/3+1, N/4+1, ...
        >
        > I've been searching for the smallest such N for each
        > successive length:
        >
        > Length N
        > ------ -
        > 1 1
        > 2 2
        > 3 12
        > 4 12720
        > 5 19440
        > 6 5516280
        > 7 5516280
        > 8 7321991040
        > 9 363500177040
        > 10 2394196081200
        > 11 3163427380990800
        >
        >
        >
        > I'm currently sieving through about 7 million candidates/second
        > while looking for the smallest such sequence of length 12.
        > I'm sure that sieve maestro Phil could beat that quite easily,
        > with his Alpha sieve programming skills, but it sure beats
        > doing trial division :-)

        Alright, just for you I ran on an x86 machine...

        <<<
        phil@nonospaz:brennen$ time ../gensv.nono brennen12.gf -s10000 -w300
        | ./brennent 12
        Verifying brennen 12
        #I ZIP: 6 primes, relation = 0 mod 30030 (expanded)
        #I SPESH: 4 primes, reduction = 215441 -> 6545
        #I SMALL@2: 4 dumb, 6 quick to 71
        #I PAUL: 190 primes <=1291
        #I WHEEL: 300 other primes <= 3643
        #I FACTOR: 2.442830e-11
        #I Flags: INFO MARKS
        BRENNEN 12 22755817971366480

        ^C
        real 24m51.336s
        user 11m51.350s
        sys 0m0.560s
        >>>

        All with 1 Fermat, so a few quick more MRs (this is a small number, I
        think that 10 MRs is enough!

        <<<
        bash-2.05a$ calc
        C-style arbitrary precision calculator (version 2.11.5t4.5)
        Calc is open software. For license details type: help copyright
        [Type "exit" to exit, or "help" for help.]

        > p=22755817971366480
        > for(i=1; i<=12; ++i) {
        >> print ptest((p+i)/i)
        >> }
        1
        1
        1
        1
        1
        1
        1
        1
        1
        1
        1
        1
        >>>

        However, it's not all peachy, the 20 minutes I spent coding the
        tester has left some bugs, as it claimed another result that only had
        11 real terms...
        ARGH - out by one error (loops from 1..12 are passed when the counter
        reaches 13!!!!)

        I did overrun quite a way before I noticed the result - shall we call
        it 8 minutes CPU time?

        Hmmm -
        10: 2394196081200
        11: 3163427380990800
        12: 22755817971366480
        Looks like either the 11 result is too high (from a moral obligation
        sense), or my 12 result is a pretender. To be honest, I didn't even
        run a quick test on <12 to see if the code works, so I've not
        verified that I find the same results as yours.
        However, I'm using the command-line/pipe version of gensv which has
        been stable for several months now, so it's only my test code that
        can have bugs (I hope).

        13 is of course going to be substantially harder.

        Phil


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      • Jon Perry
        Have you thought about extending this idea: 1)N+k is prime, N+2k is 2*{a prime}, etc... 2)N+1 is prime, N+2 is k*{a prime}, N+3 is 2*k*{a prime} 3)N+j is
        Message 3 of 6 , Apr 3, 2002
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          Have you thought about extending this idea:

          1)N+k is prime, N+2k is 2*{a prime}, etc...
          2)N+1 is prime, N+2 is k*{a prime}, N+3 is 2*k*{a prime}
          3)N+j is prime, N+2j is k*{a prime}, N+3j is 2*k*{a prime}


          Jon Perry
          perry@...
          http://www.users.globalnet.co.uk/~perry/maths
          BrainBench MVP for HTML and JavaScript
          http://www.brainbench.com


          -----Original Message-----
          From: jbrennen [mailto:jack@...]
          Sent: 27 March 2002 20:37
          To: primenumbers@yahoogroups.com
          Subject: [PrimeNumbers] Interesting prime sequence


          As an interesting little project, I've been looking for
          sequences of consecutive integers where N+1 is prime,
          N+2 is 2 times a prime, N+3 is 3 times a prime, etc.

          Another way to look at it is to find an integer N that
          maximizes the number of primes in the sequence
          N+1, N/2+1, N/3+1, N/4+1, ...

          I've been searching for the smallest such N for each
          successive length:

          Length N
          ------ -
          1 1
          2 2
          3 12
          4 12720
          5 19440
          6 5516280
          7 5516280
          8 7321991040
          9 363500177040
          10 2394196081200
          11 3163427380990800



          I'm currently sieving through about 7 million candidates/second
          while looking for the smallest such sequence of length 12.
          I'm sure that sieve maestro Phil could beat that quite easily,
          with his Alpha sieve programming skills, but it sure beats
          doing trial division :-)





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