- --- Nathan Russell <windrunner@...> wrote:
> On 2/1/07, Phil Carmody <thefatphil@...> wrote:

Correct and Correct.

> >

> > --- Shi Huang <shuangtheman@... <shuangtheman%40yahoo.com>> wrote:

> > > If the RH is false, the gap between prime P1 and the

> > > next prime P2 could be anywhere from 1 to infinitely

> > > large.

> >

> > False. Your attempt at a proof ends here.

>

> To further clarify for the original poster's benefit, of course there exist

> arbitrarily large (greater than any chosen integer) gaps following primes.

> This follows from the prime number theorem. There can be no infinite gaps,

> however, or the prime preceding the gap would be the largest (finite) prime,

> and we know that the number of primes is infinite.

However, Bertrand's postulate indicates that at prime P1 the gap must be less

than P1.

RH itself does not bound the gaps, they are already bounded. RH simply reduces

that bound (to something almost as pathetically weak as Bertrand, compared with

the empirical evidence).

This means that my comment about 3 days ago (about the RH) was in fact

incorrect.

Phil

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Have a burning question?

Go to www.Answers.yahoo.com and get answers from real people who know. - 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