> I think Dick thought you were proving that there's a

Thanks for the clarification.

> 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.

>

"Most of the fundamental ideas of science are

> > 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.

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.

>

I am talking about a law that assumes we know nothing

> > 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.

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.>

I think I am getting at the core of the primes that is

> 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.

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.

>

I think you are wrong here. Primes are unpredictable.

> > 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

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

following laws and being non-random.

> 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"?

>

as I said earlier, the law that generates primes

> Primes are of course 100% lawfully generated and

> there is no randomness.

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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http://farechase.yahoo.com/promo-generic-14795097- 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