- --- Dick <richard042@...> wrote:

>

Dick,

> > I can prove that the gap must be smaller

> > than P1xP1-P1. If P1 is 5, this means that the gap

> > between 5 and its next prime must be smaller than

> 5 x

> > 5 -5=20.

>

> Much stronger than Bertrand's postulate and a very

> sensible conjecture

> heuristically.

I am not a math specialist. So I was discouraged by

earilier replies to my post, which implies what I have

conjectured is similar to Bertrand's postulate. Could

you please explain why you think it is much stronger?

This may help me decide whether it is worthwhile to

bother with a publication of the proof.

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 wonder if you guys would consider the following a

way of proving the RH:

1. prove that primes are unpredictable, which thus

gives its random flavor.

2. prove that primes are generated by a precise law,

where no randomness is involved. The only thing that

would cause the law to behave erratically is for the

mind to make a trivial mistake, such as in generating

the series initiated by 3: 3, 6, 9, 12, 15, etc, the

mind mistakenly produces the following, 3, 6, 8, 12,

15 etc. So if the mind is like an error free computer

or machine, then no randomness at all would be

involved in generating the primes. From a physical

viewpoint, the Riemann hypothesis roughly states that

the irregularity in the distribution of primes only

comes from random noise. Here, the random noise is

the error rate of the mind.

I believe I can prove both 1 and 2. Would this be

enough to prove the RH?

Essentially, one can prove the RH using concepts from

above or from under it. The common way today is to

use advanced math concepts. If such proof is ever

possible, it would be understood by very few. The

alternative way, which may be the real way, is to

prove it using concepts that is more fundamental than

the RH. Such concept concerns directly the essence of

primes or the real definition of primes. If the

essence of primes says that prime is unpredictable,

then of course primes should give a chance appearance.

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. Simply put, besides unpredictibility,

everything else about primes is as lawful or regular

as it can possibly be, limited only by the random

noise of the mind.

So if the essence of primes is like the above, then

the RH is merely a manifestation or prediction of the

essence. The essence automatically takes care of the

RH or proves it. The RH is also a prove of the

essence. A fundamental law or essence gives many

predictions whose verification proves the law or

essence. The RH is a deduction of the prime essence

and is in turn a proof of the essence. If what the RH

means can be independently stated in other ways, the

RH is proven. That other way is to find the essence

of primes, which is the 2 statements above.

What Werner Sands said about primes being or meaning

FIRST is right on target on the essence of primes.

Thanks,

Shi

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http://smallbusiness.yahoo.com/r-index - 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