- --- Shi Huang wrote:
>

You really *can't* prove that primes cannot be predicted.

>

> > To the extent that I can understand what you're

> > saying, I

> > think that the "yin yang duality" of primes is

> > already proven.

> >

>

> If it remains unknown whether primes can be predicted,

> then at least one side of the duality is unproven. If

> one day, someone finds a formula to predict individual

> prime, then there will be no duality and primes would

> cease to be interesting. So to prove the duality, one

> must first prove that primes cannot be predicted. I

> think I may have a prove of this.

The set of primes is deterministic, and the primality of

X, or the next prime after X, or the previous prime to X --

any of those can be determined in a finite amount of time

using very simple algorithms and finite storage resources.

Now of course, it is computationally infeasible to determine

the next prime after 10^(10^100) due to physical restrictions

on computers, but proving or disproving RH won't change that.

>

The Prime Number Theorem already proved that. RH is notable

> Proving the RH will prove that primes are bound by

> laws...

> ...snip...

>

for three basic reasons (IMHO) -- it is almost certainly true (a

tremendous amount of empirical evidence points at its truth),

it bridges the gap between number theory and complex analysis,

and its proof would "tighten" the laws on prime distribution

to the point where a whole bunch of other results would become

possible. Heck, mathematicians aren't even waiting for RH

to be proven -- there are a whole bunch of published results

which are only valid if the truth of RH is assumed.

Jack - 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