AP-Prime-Generator APPG Re: possible Riemann Hypothesis proof; #137;

Yesterday I wrote:

Looking pretty good for this, this AP-Prime
Manufacturing, APPM.


Actually, I am going to need a better name since I use the term
"generations" for future sets, let me call this a AP-Prime Generator.

I looked through the math literature and it appears noone in math
history as setup this schemata of generating primes. Which seems
rather strange since Euclid's Number in Euclid's Infinitude of Primes
has been around from 2 to 3 thousand years and someone should
have devised a Generator of primes, all the primes in that time period.

So I start with {2,3}, multiply the lot add subtract
1 and enrich
my set to {2,3,5,7}. Now taking all possible
permutations of that
set, multiply the lot add subtract 1 delivers this
set {2,3,5,7,11,13,
17,19,29,31,41,43,53,71,211}

So looking very good in that the Moebius function for
the Riemann Hypothesis is bounded by the number of
permutations in each successive
generating and bound by the upper bound of that
generation, for example
the {2,3,5,7} set is bound by 2x3, 2x5, 2x7, 3x5,
3x7, 5x7, 2x3x7,
2x5x7, 3x5x7, 2x3x5x7

And bound by the range, for example the range of
{2,3,5,7} is 210. The
range of {2,3} is 6.

So it looks ideal for the Moebius function
equivalency of the Riemann Hypothesis.

But the question remains, is this APPM going to
generate all the primes?
This maybe a difficult proof. The way I see it, is
that APPM is the reverse
of the Sieve of Eratosthenes. And the sieve needs no
proof and guarantees
all the primes. So I wonder, is my APPM just the
Sieve only slightly different
where instead of discarding composites, I retrieve
the primes within the composites.

So has anyone proven that APPM includes all the
Primes? The APPM may go under a different name.


Proof of the Sieve of Eratosthenes:

(1) We need to prove that this Sieve captures all the primes
(2) We list all the positive integers
(3) Primes are a subset of all the positive integers
(4) The Sieve merely determines composites and discards them.
(5) The Sieve leaves behind the primes
(6) Since the primes are a subset of all the positive integers, the
Sieve of Eratosthenes thus is a set of all the primes.

Now I need a proof of the AP-Prime-Generator APPG that
it captures all the primes.

This is probably not going to be easy.

To prove the Riemann Hypothesis with using APPG, it is easy to get
whatever I want for the Moebius Function in terms of bounds on the
function since I can play around and tinker around with the generator
output and input. I can adjust alot of parameters of the generator to
make whatever ideal bounds on the Moebius function. So that part of
the Riemann Hypothesis is going to be easy. And looking at the rest
of the Riemann Hypothesis of its Zeta function and zeroes thereof, why
was the RH so extremely difficult to prove? Apparently it was not because
of the bounds and conditions of the Zeta function or Moebius function,
but rather, the hardship and extreme difficulty of proving the RH was bundled up in this AP-Prime-Generator, in whether those generators
produce ALL the primes. We see the Sieve of Eratosthenes produces all the
primes because we are given them from the start. But here, with the APPG
and the Riemann Hypothesis, we do not know if the primes are all generated. In RH, the primes are given, so then it all is a question not of
all primes but whether all the zeroes are well behaved. With the Moebius,
I can get the boundaries easily enough by adjusting the Generators, so the
proof then hinges only on whether all the primes are produced.

Now I see that "2" is critical in this APPG. For the Euclid's Number of +, - 1
generator I can start with simply the singleton set of {1} and generate
the prime 2. Now I have the set {2} which yields in the next generation the
set {2,3}.

Now I can adjust the generator by saying I want only pairwise Euclid Numbers so that the set {2,3} works on 2x2, 2x3, 3x3 yielding the new
primes of 5,7.

As for the Generators of Euclid Numbers such as +,-2 it seems to work
provided I start with a set such as this {2,3,5}. And it seems to work if
I have Euclid Numbers of +-3, then 4, then 5 etc etc provided I start with
a robust enough initial set.

So at this moment in time, after some calculations and playing around,
I am confident that those generators all, yield all the primes. But I am
still at a loss as to why they produce all the primes. They look like a
Reversal of the Sieve of Eratosthenes.

So if I can prove that the APPG produces all the primes and that this
schemata can easily regulate the bounds of the Moebius function, then
I think I have a proof of the Riemann Hypothesis.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
.

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