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Factor formulas

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  • Kermit Rose
    If z = 1 + 2 z1 + 4 z2 + 8 where z1 and z2 are variables that take on only the values of 0 or 1, then 8
    Message 1 of 2 , Oct 7, 2007
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      If z = 1 + 2 z1 + 4 z2 + 8
      where z1 and z2 are variables that take on only the values of 0 or 1,

      then

      8 < z < 16
      z is odd

      if z1 is different than z2, then z is prime,
      and

      if z is composite, then

      z = 3 * (3 + 2 z1)


      The algorithm I used to derive this prime number test, and factoring
      formula will apply to higher powers of 2.

      Derivation of the prime formula tests and the factoring formula is more
      difficult than factoring any particular
      odd z in the specified range.

      And I do not know whether or not the factoring formula derived for
      higher powers of 2 will provide a quick way of
      factoring arbitrary odd integers in the given range.

      But I do expect to be able to derive a formula that will describe the
      factors of every odd integer between consecutive powers of 2.

      The derivation is tedious, and I'm still working on the derivation of
      the prime number test and factoring formula for odd integers between
      16 and 32.

      Does anyone wish to work with me to derive these factoring formulas and
      prime number tests formula?

      Kermit < kermit@... >
    • Paul Leyland
      ... Trivial. ... Trivial. ... Also trivial. All odd composites
      Message 2 of 2 , Oct 8, 2007
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        On Sun, 2007-10-07 at 23:39, Kermit Rose wrote:
        > If z = 1 + 2 z1 + 4 z2 + 8
        > where z1 and z2 are variables that take on only the values of 0 or 1,
        >
        > then
        >
        > 8 < z < 16
        Trivial.

        > z is odd

        Trivial.

        > if z1 is different than z2, then z is prime,
        > and
        >
        > if z is composite, then
        >
        > z = 3 * (3 + 2 z1)

        Also trivial. All odd composites < 16 are multiples of 3 and 5.

        > The algorithm I used to derive this prime number test, and factoring
        > formula will apply to higher powers of 2.

        Quite possibly true, but in the absence of an explanation of your
        algorithm, impossible to verify.. Is your algorithm more efficient than
        the well-known alternatives?

        > Derivation of the prime formula tests and the factoring formula is
        > more
        > difficult than factoring any particular
        > odd z in the specified range.
        >
        > And I do not know whether or not the factoring formula derived for
        > higher powers of 2 will provide a quick way of
        > factoring arbitrary odd integers in the given range.
        >
        > But I do expect to be able to derive a formula that will describe the
        > factors of every odd integer between consecutive powers of 2.
        >
        > The derivation is tedious, and I'm still working on the derivation of
        > the prime number test and factoring formula for odd integers between
        > 16 and 32.
        >
        > Does anyone wish to work with me to derive these factoring formulas
        > and
        > prime number tests formula?

        Not I.


        Paul



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