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Re: sufficent proof for primes ?

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  • djbroadhurst
    ... If p is an odd prime, then the group (Z/Zp)* of integers from 1 to p-1 is cyclic under multiplication modulo p. If n is a square-free odd composite
    Message 1 of 50 , Aug 9, 2010
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      --- In primenumbers@yahoogroups.com,
      "bhelmes_1" <bhelmes@...> wrote:

      > The description of a cyclic group is borrowed from the
      > cyclic group of natural numbers.

      If p is an odd prime, then the group (Z/Zp)*
      of integers from 1 to p-1 is cyclic under
      multiplication modulo p.

      If n is a square-free odd composite integer, then
      the group (Z/Zn)* of integers from 1 to n-1 that
      are coprime to n is _not_ cyclic.

      > You are very fast in judging the proof.

      That was because it contained obvious "cyclic" nonsense.

      > I did not wrote that f and g must be primes

      That may be because you do not understand cyclic groups.

      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 4:14 AM
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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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