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Re: it works!

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  • Alan Eliasen
    ... Well, it fails on even numbers, for one. Additional failures occur a bit later on odd numbers: x y Error at 1729 2 4 Error at 2821 2
    Message 1 of 5 , Feb 5, 2008
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      leavemsg1 wrote:

      > the following program tries to determine the primality of a number by
      > combining two tests into one, avoiding the usual pseudo-prime situ-
      > ation:
      >
      > 10 cls
      > 20 input N
      > 30 NN= int((N-1)/2)
      > 40 X= (2^N-3^NN-(N-1)) mod N
      > 50 Y = N mod 5
      > 60 print N;X
      > 70 if (X=2 or X=4) and Y<>0 then print "prime"
      > 80 goto 10
      >
      > it inevitably misses the 2, 3, and 5, but catches all the rest as far
      > as I can test it.
      >
      > my computer is somewhat outdated, but it told me that there were 168
      > primes before the number 1001, when I used this code inside a for-
      > loop.
      >
      > i am aware of the law of small numbers, but can anyone realistically
      > find a flaw or counter-example in this unproven technique ???

      Well, it fails on even numbers, for one.

      Additional failures occur a bit later on odd numbers:

      x y
      Error at 1729 2 4
      Error at 2821 2 1
      Error at 7921 2 1
      Error at 8911 4 1
      Error at 13699 4 4
      Error at 15841 2 1
      Error at 18721 2 1
      Error at 29341 2 1
      Error at 31621 2 1

      Sorry.

      --
      Alan Eliasen | "Furious activity is no substitute
      eliasen@... | for understanding."
      http://futureboy.us/ | --H.H. Williams
    • Alan Eliasen
      ... Hopefully for Bertrand, he didn t say that. If I m supposed to be finding counterexamples and errors today, that single sentence contains a treasure
      Message 2 of 5 , Feb 6, 2008
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        Danny Fleming wrote:
        > Bertrand's postulate proved that there are at least two primes from x
        > to 2x for any integer.

        Hopefully for Bertrand, he didn't say that. If I'm supposed to be
        finding counterexamples and errors today, that single sentence contains
        a treasure trove.

        You may also wish to mention how you believe it related to the
        messages that you replied to.

        --
        Alan Eliasen | "Furious activity is no substitute
        eliasen@... | for understanding."
        http://futureboy.us/ | --H.H. Williams
      • Jacques Tramu
        Assuming N odd, First counter-examples with their prime factors : 1729 { 7, 13, 19} 2821 { 7, 13, 31} 7921 { 89} 8911 { 7, 19, 67} 13699 { 7, 19,
        Message 3 of 5 , Feb 6, 2008
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          Assuming N odd, First counter-examples with their prime factors :

          1729 { 7, 13, 19}
          2821 { 7, 13, 31}
          7921 { 89}
          8911 { 7, 19, 67}
          13699 { 7, 19, 103}
          15841 { 7, 31, 73}
          18721 { 97, 193}
          29341 { 13, 37, 61}
          31621 { 103, 307}
          32761 { 181}
          41041 { 7, 11, 13, 41}
          46657 { 13, 37, 97}
          49141 { 157, 313}
          52633 { 7, 73, 103}
          63973 { 7, 13, 19, 37}
          64177 { 29, 2213}
          75361 { 11, 13, 17, 31}
          83333 { 167, 499}
          93961 { 7, 31, 433}
          115921 { 13, 37, 241}
          126217 { 7, 13, 19, 73}
          129481 { 11, 79, 149}
          162401 { 17, 41, 233}
          .......
          if( N < 10 000 000) whe have 123 failures.

          Regards,
          JT


          >the following program tries to determine the primality of a number by
          >combining two tests into one, avoiding the usual pseudo-prime situ-
          >ation:

          10 cls
          20 input N
          30 NN= int((N-1)/2)
          40 X= (2^N-3^NN-(N-1)) mod N
          50 Y = N mod 5
          60 print N;X
          70 if (X=2 or X=4) and Y<>0 then print "prime"
          80 goto 10

          >it inevitably misses the 2, 3, and 5, but catches all the rest as far
          >as I can test it.

          >my computer is somewhat outdated, but it told me that there were 168
          >primes before the number 1001, when I used this code inside a for-
          >loop.

          >i am aware of the law of small numbers, but can anyone realistically
          <find a flaw or counter-example in this unproven technique ???

          >Bill B

          ---------------------------------------
          http://www.echolalie.com
          ---------------------------------------





          [Non-text portions of this message have been removed]
        • Peter Kosinar
          Hello Bill, ... Slightly less programatic description of the test: If both 1) N must not be divisible by 5, and 2) 2^N - 3^((N-1)/2) = 1 or 3 (mod N), then N
          Message 4 of 5 , Feb 7, 2008
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            Hello Bill,

            > the following program tries to determine the primality of a number by
            > combining two tests into one, avoiding the usual pseudo-prime situ-
            > ation:

            Slightly less programatic description of the test: If both
            1) N must not be divisible by 5, and
            2) 2^N - 3^((N-1)/2) = 1 or 3 (mod N),
            then N is declared prime.

            Clearly, this test is surely not stronger (although it is likely strictly
            weaker) than a combination of
            1) Divisibility by 5 test,
            2) Base 2 Fermat test and
            3) Base 3 Euler test
            while having about the same time-complexity as these three together (in
            fact, performing the step 3 only if step 2 succeeds can improve the
            actual speed).

            Combining the tests the way you did is usually not a good idea, as you're
            losing some potentially useful information. In particular, instead of
            checking 2^N = 2 (mod N) and 3^((N-1)/2) = +/-1 mod N, you're just
            checking if their difference is equal to something -- which can, unless
            proven otherwise, happen a lot more often.

            Peter

            --
            [Name] Peter Kosinar [Quote] 2B | ~2B = exp(i*PI) [ICQ] 134813278
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