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Interesting prime sequence

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  • jbrennen
    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,
    Message 1 of 6 , Mar 27, 2002
      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 :-)
    • Jon Perry
      ... should obviously be highlighted, and postulated upon such that an infinite number of such pairs exist. Jon Perry perry@globalnet.co.uk
      Message 2 of 6 , Mar 27, 2002
        > 6 5516280
        > 7 5516280

        should obviously be highlighted, and postulated upon such that an infinite
        number of such pairs exist.

        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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        The Prime Pages : http://www.primepages.org



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      • Jud McCranie
        ... You need to send this to Sloane. +---------------------------------------------------------+ ...
        Message 3 of 6 , Mar 27, 2002
          At 08:36 PM 3/27/2002 +0000, jbrennen wrote:

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


          You need to send this to Sloane.



          +---------------------------------------------------------+
          | Jud McCranie |
          | |
          | Programming Achieved with Structure, Clarity, And Logic |
          +---------------------------------------------------------+
        • 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 4 of 6 , Mar 27, 2002
            --- 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 5 of 6 , Mar 28, 2002
              --- 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 6 of 6 , Apr 3, 2002
                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 :-)





                Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
                The Prime Pages : http://www.primepages.org



                Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
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