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Re: "wriggly" probable primes

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  • Aldrich
    ... If a different mode of analysis is used there is no wriggly problem. If A = 8635073539 then A appears twice on A = 5x^2 + 5xy + y^2 at x = 14353, y =
    Message 1 of 50 , Sep 7, 2010
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      --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
      >
      > Here is an example with a 42nd root of unity:
      >
      > p=8635073539; a=1342131859; print(znorder(Mod(a,p)));
      > 42
      >
      > It fools these test:
      >
      > > 1. p is an odd positive integer,
      > > 2. a is an integer with kronecker(a,p) = -1,
      > > 3. a^((p-1)/2) = -1 mod p,
      > > 5. Mod(1+x,x^2-a)^p = 1-x mod p,

      > David
      >

      If a different mode of analysis is used there is no
      "wriggly" problem. If A = 8635073539 then A appears
      twice on A = 5x^2 + 5xy + y^2 at x = 14353, y = 58418
      and at x = 28701, y = 26557, and is therefore composite.

      Aldrich Stevens
    • 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
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        --- 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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