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Re: "wriggly" probable primes

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  • djbroadhurst
    ... Exercise: Find two pairs of positive integers (x,y) such that 4065702994722252685573484796054334194691713593576645739409115721859519 = 5x^2 + 5xy + y^2
    Message 1 of 50 , Sep 8, 2010
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      --- In primenumbers@yahoogroups.com,
      "Aldrich" <aldrich617@...> wrote:

      > appears twice on A = 5x^2 + 5xy + y^2
      > and is therefore composite

      Exercise: Find two pairs of positive integers (x,y) such that
      4065702994722252685573484796054334194691713593576645739409115721859519
      = 5x^2 + 5xy + y^2

      Comment: Pari-GP's "qfbsolve" enables a solution in two minutes.
      Devotees of "issquare", like Aldrich, may take considerably longer.

      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
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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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