## Re: [PrimeNumbers] Re: proving the Riemann hypothesis

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• ... Thanks for the clarification. ... Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language
Message 1 of 40 , Feb 2, 2007
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> I think Dick thought you were proving that there's a
> prime between p^2
> - p and p^2 + p.
> I think the rest of us thought you were proving that
> there's a prime
> between p and p^2 - p.

Thanks for the clarification.
>
> > I now also realized that my conjecture is not
> enough
> > to prove the RH. It mearly says that the gap is
> not
> > completely random but it does not say that the gap
> is
> > as non-random as possble which is what the RH
> means.
>
> I think that these kinds of very rough paraphrases
> are not very useful.

"Most of the fundamental ideas of science are
essentially simple, and may, as a rule, be expressed
in a language comprehensible to everyone."
Albert Einstein (1879-1955)

If we cannot prove the RH's exact statement, we may
try to prove its simple and layperson statement. The
RH is widely regarded as the most fundamental problem
of math, if not mankind. Its basic idea should be
comprehensible to everyone. If it is wrong, it should
strike everyone as unreasonable. The thing about
primes that bugs people and makes it uniquely
interesting is the two opposite yin and yang facts as
commented by Don Ziger: being both random and
non-random. Neither facts are however proven and are
only seemingly so. The RH supports the non-random
aspect of primes. If primes lacks randomness, its
non-randomness and in turn RH would be much less
interesting. It is the seemingly impossible unity of
two complete opposites that makes the prime so
interesting.

If the generation of primes, by the sieve method or
the division method, can be chacterized or proven as
non-random, we have at best proven only half of the
picture. We must also prove that prime is random.
What random means depends on how you want to prove it.
The simplest is to prove that primes are
unpredictable and no one has been able to do this.

>
> > 2. prove that primes are generated by a precise
> law,
> > where no randomness is involved.
> Of course there's no randomness involved -- a number
> is either prime
> or it isn't, and we can compute that.

I am talking about a law that assumes we know nothing
about the divisibility of primes. We know only that
prime has one property which makes it both
unpredictable as well as lawful or regular. The
divisibility property of primes may make prime lawful
but does not make or has not been proven to make prime
unpredictable. It is easy to prove either the yin or
yang alone, but it is hard to prove both yin and yang
with the same stone. We need to find one property or
essence of primes that takes care of both yin and
yang.
>
> The question is in what ways the sequence of primes
> "looks like" a
> random sequence, by which we mean that properties
> like the length of
> the shortest gap in a given range of numbers, or the
> longest gap, or
> the percent prime, might be similar to a
> "probabilistically-generated
> prime-like" sequence.
>
> > I believe I can prove both 1 and 2. Would this be
> > enough to prove the RH?
>
> No.

I think I am getting at the core of the primes that is
even more fundamental than the RH. The core is the
yin and yang unity of random and non-random. we need
to find a prove or reason for that unity. The RH is
merely one of the two poles of the bipolar unity.
Proving the RH alone would still not be enough to
explain the unity. We still need to prove the
randomness, which we can accomplish by proving
unpredicatibility. So, I suggest a conjecture that
says that there is a law that can generate individual
prime precisely (non-random/lawfulness) but also
demands that prime is unpredictable
(random/lawlessness). Proving this conjecture or
finding this law, in my view, takes care of the
yin-yang duality that makse prime so interesting, and
is in turn much more important than proving the RH.

>
> > If the
> > essence of primes says that prime is
> unpredictable,
> > then of course primes should give a chance
> appearance.
>
> What makes primes so interesting is that they are
> predictable

I think you are wrong here. Primes are unpredictable.
The total numbers under a number may be but not
individual prime. If you cannot predict individual
prime, then prime is unpredictable but this needs to
be proven either false or true.

What makes prime so interesting is the unity of random
and non-random. But you confused non-random with
predictability of total number of primes under a
number. If random means unpredictability by way of a
law or lawlessness, then its opposite must mean
precise generation by a law or lawfulness. the same
law that generates primes lawfully must in the mean
time says that primes are unpredictable or random. No
such law exists today. The sieve method or the
division method may qualify as laws in precisely
generating primes but they do not say or prove that
primes must be random or unpredictable.

Indeed, any law or conjecture that proves that primes
are unpredictable would be a major breakthrough. It
would save many people from wasting time trying to
find a formula to predict individual prime.

and yet
> still in many ways give a chance appearance: the
> sequence of primes
> has many, many properties in common with a
> probabilistically-generated
> sequence.
>
> > If the essence also says that primes are 100%
> > lawfully generated where randomness is completely
> > ruled out, then of course, primes would show
> extreme
> > regularity.
>
> But what exactly do you mean by "regularity"?

following laws and being non-random.

>
> Primes are of course 100% lawfully generated and
> there is no randomness.

as I said earlier, the law that generates primes
precisely must at the same time entails randomness to
primes. No such law exists.

We need to find a prime generating law that alone
produces the random/lawlessness and
non-random/lawfulness property of primes, which is the
core of what makes prime so interesting, much more so
than the RH which is merely one manifestation of the
core. The RH, if proven true, proves the
non-randomness of primes. but for that to be
interesting, we must prove that primes are also
random. The fact that people are still looking for
formula to predict individual primes is plain evidence
that no one has proven that primes are indeed random
or unpredictable. They just seem to be random or show
many properties of randomness. But we still need a
proof to be sure.

The universe seems to be randomly created, and most
scientists believe it is. But there is no proof. The
universe also seem to follow laws. So the universe
(and all physical reality) is the closest thing to the
primes, display a unity of random and non-random. It
is easy to have laws that only describe the random
aspect. It is also easy to have laws that only
describe the non-random aspect. But what we need is a
single law that explains or proves both the random and
non-random aspects. Indeed, solving the prime puzzle
may automatically take care of the deepest puzzle of
the universe.

Shi

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• For example 2 adjacent gaps cannot be equal if they aren t multiple of 6. For example the gap between 2 pairs of twins is at least 4. For example each prime
Message 40 of 40 , Feb 7, 2007
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For example 2 adjacent gaps cannot be equal if they aren't multiple of
6. For example the gap between 2 pairs of twins is at least 4. For
example each prime number has the form 2n+/-1, 3n+/-1, 4n+/-1, 6n+/-1.
Each pair of twins has the form 12n+-1, there are approximate formulas
for the nth prime and the number of primes < x and so on. You cannot
call all this random ore unpredictable. Of course the prime numbers are
distributed as regularly as possible, that's a tautology. In
mathematics everything is as regular as possible. Is pi random? Build
P=2,357111317192329, and you have the same case as pi. Consider the
primes to be an irrational number, and there are no problems. If you
mean there is no formula f(n) which produces primes for each n, then
you are right. In this sense primes are random. (I am not quite sure 
there is a formula p=[k^n^3] (H.W.Mills) which is said to produce only
prime numbers). If you define "formula" as an algorithm, as a
calculation instruction such as the sieve of Eratosthenes, then the
primes are not random but simply what they are. Perhaps the compound
numbers are random? Or are they only non-transparently complicated?

Werner
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