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Re: "wriggly" probable primes

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  • djbroadhurst
    Here is an example with a 42nd root of unity: p=8635073539; a=1342131859; print(znorder(Mod(a,p))); 42 ... since the gcd extracts a factor:
    Message 1 of 50 , Sep 2, 2010
      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,

      but not my gremlin-trap:

      > 4. gcd(a^6-1,p) = 1,

      since the gcd extracts a factor:

      print(gcd(a^6-1,p));
      1593481

      Explanation: p = 5419*1593481.
      znorder(Mod(a,5419)) is 42, but
      znorder(Mod(a,1593481)) is 6.

      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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