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fact or fiction

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  • Bill Bouris
    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
    Message 1 of 4 , Feb 1, 2010
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      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
    • Jack Brennen
      Try N = 25326001. Try N = 3215031751. Then fix the test to handle those correctly. :)
      Message 2 of 4 , Feb 1, 2010
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        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 3 of 4 , Feb 1, 2010
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          --- 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 4 of 4 , Feb 2, 2010
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            --- 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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