- On 2/2/07, Shi Huang <shuangtheman@...> wrote:
>

I think Dick thought you were proving that there's a prime between p^2

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

> > > 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.

>

> Dick,

>

> 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?

- 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.

> I now also realized that my conjecture is not enough

I think that these kinds of very rough paraphrases are not very useful.

> 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.

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

Of course there's no randomness involved -- a number is either prime

> where no randomness is involved.

or it isn't, and we can compute that.

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

No.

> enough to prove the RH?

> If the

What makes primes so interesting is that they are predictable and yet

> essence of primes says that prime is unpredictable,

> then of course primes should give a chance appearance.

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%

But what exactly do you mean by "regularity"?

> lawfully generated where randomness is completely

> ruled out, then of course, primes would show extreme

> regularity.

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

--Joshua Zucker - 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