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Re: sufficient proof for primes ?

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  • djbroadhurst
    ... [4] is meaningless, as it stands. You should write a double mod: 4. (1+x)^p = 1-x mod(x^2-a,p) ... There is no reason whatsoever to believe that [1] to [4]
    Message 1 of 50 , Sep 29, 2011
      --- In primenumbers@yahoogroups.com,
      "bhelmes_1" <bhelmes@...> wrote:

      > 1. let a jacobi (a, p)=-1
      > 2. let a^(p-1)/2 = -1 mod p
      > 3. a^6 =/= 1 mod p
      > 4. (1+sqrt (a))^p = 1-sqrt (a)

      [4] is meaningless, as it stands.
      You should write a double mod:

      4. (1+x)^p = 1-x mod(x^2-a,p)

      > 1. Is it possible that there are other exceptions

      There is no reason whatsoever to believe that
      [1] to [4] establish that p is prime. Morevoer,
      some folk believe that, for every epsilon > 0,
      the number of pseudoprimes less than x may
      exceed x^(1-epsilon), for /sufficiently/ large x.

      > 2....
      > there is a cyclic order ...

      > 3....
      > there is a cyclic order ...

      The group of units (Z/nZ)* is /not/ cyclic
      if n has at least two distinct odd prime fators.

      David
    • djbroadhurst
      ... [4] is meaningless, as it stands. You should write a double mod: 4. (1+x)^p = 1-x mod(x^2-a,p) ... There is no reason whatsoever to believe that [1] to [4]
      Message 50 of 50 , Sep 29, 2011
        --- In primenumbers@yahoogroups.com,
        "bhelmes_1" <bhelmes@...> wrote:

        > 1. let a jacobi (a, p)=-1
        > 2. let a^(p-1)/2 = -1 mod p
        > 3. a^6 =/= 1 mod p
        > 4. (1+sqrt (a))^p = 1-sqrt (a)

        [4] is meaningless, as it stands.
        You should write a double mod:

        4. (1+x)^p = 1-x mod(x^2-a,p)

        > 1. Is it possible that there are other exceptions

        There is no reason whatsoever to believe that
        [1] to [4] establish that p is prime. Morevoer,
        some folk believe that, for every epsilon > 0,
        the number of pseudoprimes less than x may
        exceed x^(1-epsilon), for /sufficiently/ large x.

        > 2....
        > there is a cyclic order ...

        > 3....
        > there is a cyclic order ...

        The group of units (Z/nZ)* is /not/ cyclic
        if n has at least two distinct odd prime fators.

        David
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