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Re: [PrimeNumbers] fact or fiction

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  • Jack Brennen
    Try N = 25326001. Try N = 3215031751. Then fix the test to handle those correctly. :)
    Message 1 of 4 , Feb 1, 2010
      Try N = 25326001.
      Try N = 3215031751.

      Then fix the test to handle those correctly. :)


      Bill Bouris wrote:
      > Here's how I want to define the magnum 357 test!
      >
      > take N; if 'N' passes the 2-PRP test, then...
      >
      > if 3, 5, or 7 divides 'N', then 'N' is composite... and we are done.
      >
      > (or) if either ALL or NONE of them divide 'N-1', then a simple 3-PRP
      > will conclude that 'N' is composite, without a doubt.
      >
      > (or) if either one or two of them divide 'N-1', then a test can be
      > devised to determine whether 'N' is prime or composite, no doubts.
      >
      > say only 3 doesn't, then either 3^((N-1)/5) mod N or 3^((N-1)/7) mod
      > N will prove that N is prime or composite when followed by a second
      > test of either (result)^5 mod N or (result)^7 mod N, respectively.
      >
      > I believe that the {(result)^(power) mod N} <> abs((power)-3) will
      > confirm that 'N' is composite, otherwise 'N' will be prime.
      >
      > There's probably not enough theory to deny or support this conclusion.
      >
      > Either I'm right or wrong... I can accept it... non-spammingly! I'm also
      > watching for the Troll-watcher(DBr)... lol
      >
      > Bill
      >
      >
      >
      >
      >
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    • djbroadhurst
      ... It takes less than 2 seconds to prove that there are at least 124880 counterexamples with N
      Message 2 of 4 , Feb 1, 2010
        --- In primenumbers@yahoogroups.com,
        Bill Bouris <leavemsg1@...> wrote:

        > if 'N' passes the 2-PRP test, then...
        > if 3, 5, or 7 divides 'N', then 'N' is composite...
        > (or) if either ALL or NONE of them divide 'N-1',
        > then a simple 3-PRP will conclude that 'N' is composite, 
        > without a doubt.

        It takes less than 2 seconds to prove that there are
        at least 124880 counterexamples with N < 10^15. See:
        http://physics.open.ac.uk/~dbroadhu/cert/ct124880.zip

        It is likely that there are *precisely*
        124880 counterexamples with N < 10^15

        David
      • djbroadhurst
        ... In less than 20 seconds, I found 993305 counterexamples in tables from William Galway and Richard Pinch. There are 319226 numbers N
        Message 3 of 4 , Feb 2, 2010
          --- In primenumbers@yahoogroups.com,
          Bill Bouris <leavemsg1@...> wrote:

          > if 'N' passes the 2-PRP test, then...
          > if 3, 5, or 7 divides 'N', then 'N' is composite...
          > (or) if either ALL or NONE of them divide 'N-1',
          > then a simple 3-PRP will conclude that 'N' is composite,
          > without a doubt.

          In less than 20 seconds, I found 993305 counterexamples
          in tables from William Galway and Richard Pinch.

          There are 319226 numbers N < 10^15 that are 2-PSP and 3-PSP.
          Of these, 287672 are coprime to 3*5*7.
          Of these, 119450 have N-1 divisible by 3*5*7
          and another 5430 have N-1 coprime to 3*5*7
          giving us 124880 counterexamples below 10^15.

          There are 1296432 Carmichael numbers C between 10^15 and 10^18.
          Of these, 1131205 are coprime to 3*5*7.
          Of these, 868340 have C-1 divisible by 3*5*7
          and another 85 have C-1 coprime to 3*5*7
          giving us 868425 additional counterexamples.

          Tally and timing from Pari-GP:
          124880 + 868425 = 993305 counterexamples in 19320 milliseconds

          David
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