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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
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@... >
• ... Trivial. ... Trivial. ... Also trivial. All odd composites
Message 1 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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