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From: Archimedes Plutonium on 18 Jul 2010 03:02


Archimedes Plutonium wrote:
> Archimedes Plutonium wrote:
> > I am looking for the best Riemann Hypothesis equivalent statement to
> > tie in the Indirect Euclid Infinitude of Primes proof method. By
> > correcting that flaw of logic that both P-1
> > and P+1 are necessarily prime, yielding the infinitude of Twin Primes,
> > I suspect is a
> > key to proving the Riemann Hypothesis RH.
> >
> > So I looked for equivalent RH statements:
> > --- quoting Wikipedia in part ---
> > Riemann's explicit formula for the number of primes less than a given
> > number in terms of a sum over the zeros of the Riemann zeta function
> > says that the magnitude of the oscillations of primes around their
> > expected position is controlled by the real parts of the zeros of the
> > zeta function. In particular the error term in the prime number
> > theorem is closely related to the position of the zeros: for example,
> > the supremum of real parts of the zeros is the infimum of numbers β
> > such that the error is O(xβ) (Ingham 1932).
> >
> >
> > Von Koch (1901) proved that the Riemann hypothesis is equivalent to
> > the "best possible" bound for the error of the prime number theorem.
> >
> >
> > A precise version of Koch's result, due to Schoenfeld (1976), says
> > that the Riemann hypothesis is equivalent to. . .
> >
> > --- end quoting ---
>
> Ingham, Von Koch, and Schoenfeld and others bespeak of the Riemann
> Hypothesis
> as the most efficient placing of primes in a prime distribution. As if
> efficiency and accuracy
> of placement of primes is what the Riemann Hypothesis is all about.
>
>
> >
> > Let me try to give an equivalent RH statement myself.
> >
> > It is already proven, I think it was Chebychev, that between n and 2n
> > always exists another prime.
> >
> > So, let me focus on n+1 and 2n-1
> >
> > We have:
> >
> > for 2, 2+1 = 3 and 4-1 = 3
> >
> > for 3, 3+1=4 and 6-1=5
> >
> > for 4, 4+1 =5 and 8-1=7
> >
> > for 5, 5+1=6 and 10-1=9
> >
> > etc etc
> >
> > Now, instead of Riemann getting involved with the Complex Number
> > Plane, how about a
> > Riemann Hypothesis more down to Earth. How about a Riemann Hypothesis
> > with just the plain old Natural Numbers since we find billions and
> > zillions of equivalent statements, but
> > never the most simple statement.
> >
> > So let me proffer my own equivalent statement of the Riemann
> > Hypothesis since the one
> > thing that RH can never get away from is the distribution of prime
> > numbers.
> >
> > Archimedes Plutonium's equivalent statement of the Riemann Hypothesis:
> > The RH, if true says that as n becomes large, very large that both n+1
> > and 2n-1
> > are both prime numbers. If that is true, then a proof of that RH
> > equivalent is easily
> > begot from the Euclid Infinitude of Primes proof Indirect method for
> > it makes
> > n+1 and 2n-1 necessarily new primes as n goes to infinity.
>
>
> Then this equivalent statement to the RH by myself is not efficient
> and accurate enough.
>
> I should have said that the RH equivalent is such that n-1, n+1 and
> 2n-1, 2n+1, all four
> of those numbers are necessarily prime as n tends to infinity.
>
> An example of that is n=30 so that n-1 =29 and n+1=31, and
> 2n-1=60--1=59 and 2n+1=
> 60+1=61 are all four prime numbers. So that would be a Maximum density
> of primes
> given n goes to infinity.
>
> It is where the Infinitude of Primes proof conjoins with the Riemann
> Hypothesis, and the
> proof of this RH is simply a Indirect Method with Mathematical
> Induction that yields four
> Euclid Numbers, all four of which are necessarily prime numbers.
>
>
>
> >
> > Now I am curious since I define with precision the finite-number
> > versus the infinite-number
> > as the boundary at 10^500. So I am curious as to whether 10^500 (+1)
> > is a prime number
> > and its associate of 2x(10^500) -1. If not, then let us chose as the
> > boundary where n+1
> > and 2n-1 in the region of 10^500 are both prime numbers. So that
> > mathematics does share
> > a input into the selection of the boundary between finite and infinite-
> > number.
> >
> > Perhaps a major reason the RH was never proven or steered into a
> > correct path to prove it, was that it was too much cloaked in the
> > Complex Number Plane and if someone had retrieved it out of that
> > cloaking, would have seen it in its more basic form that n+1 and 2n-1
> > are both
> > primes when n tends to infinity. They may not have realized that a
> > simple tinker to fix the logic flaw of Euclid IP indirect, but at
> > least they would have made RH more understandable.
> > Mathematicians are like artists, once they paint legs on a snake, they
> > refuse to remove the legs and rather increase the complexity.
> >
>
> Now Physics is the king of sciences and mathematics is only a room, a
> tiny
> room in the house of physics. And Physics would define the boundary
> between
> finite number versus infinite-number and it would be the largest
> Planck unit
> which is the Coulomb Interactions in element 100 of about 10^500. But
> here is where
> mathematics has a "say at the table". Since the RH of above would have
> four primes
> at n, 2n, the question is does 10^500 plus and minus 1 yield twin
> primes and does
> 2x10^500 plus and minus 1 yield twin primes? If so, then we assuredly
> take 10^500 as
> the boundary between finite-number versus infinite-number. Or if there
> is another large
> number in the vicinity of 10^500 that yields those four primes.
>
> Carbon in me, carbon of plutonium, fill me with life anew, that I may
> love what thou dost
> love. Oxygen in me, oxygen of plutonium. .
>

I suppose there is even more symmetry to add to the above. In my
example of
30 with twin primes astride 30 and then 2n as 60 with twin primes
astride, which
is very symmetrical because the interval has two primes but also the
interval
below 30 and the interval above 60 already have their primes set to
go.

But there is additional symmetry. Keep in mind I am striving for a
maximum Riemann Hypothesis as to the maximum efficiency of primes. So
in the interval 30 to 60 there is
a possibility of having the midpoint be a prime, in this case 45 is
not prime, but in another
case of n to 2n have (n + 2n)/2 be a prime.

Off hand I cannot provide an example of that. And should the boundary
marker of finite-number
with infinite-numbers such as 10^500 be an example of that situation?

I think not, because the Infinitude of Primes Proof, indirect does not
venture into whether the
midpoint is prime.

The whole idea of this most simple and elegant statement for the
equivalent of the Riemann Hypothesis is to measure RH against the full
potential of the Indirect IP proof. The full potential is that we end
up with four new necessarily primes as Euclid Numbers of the twin
primes on each end node of n and 2n. Euclid IP Indirect is silent
about a midpoint prime.
So I think I have captured the full extent of RH with Euclid IP.

If I am correct, then the Euclid Infinitude of Primes Proof, Indirect
proves the Riemann Hypothesis as true.

Now why would RH escape that attention? Probably because noone saw the
mistake in
Euclid IP Indirect and how much it influences the rest of mathematics.

But that leaves me in a funny position, for it leaves me only with FLT
to complain about that it has no proof until a precision definition of
finite versus infinite-number is given as 10^500 or
thereabouts. When I had RH to include with FLT, I felt a bit better of
a stronger case. But now it looks as though FLT is the only one
standing that demands and crys out for a precision boundary definition
of finite versus infinite. But maybe, just maybe the nonmath
conjecture of engineering called the NP conjecture is mincemeat once
10^500 is made the boundary.

Archimedes Plutonium
http://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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