- Le 2013-04-02 21:34, viva8698 a écrit :
> Recently we had a great discussion during a meeting with colleague

There is no such computationally efficient *formula*.

> matematicians and we opened a theme that seems to get quite an

> interesting intellectual quiz:

>

> What would be the consequences if Mr. X would have a formula to

> calculate from a given prime the whole ordered series of the

> subsequent primes one after the other?

However there are _algorithms_, with obvious drawbacks.

It's not what most mathematicians want, hope or dream of,

but Mathematics is not created to please us either. - Lets have a vote everybody on one question then:

Do we need a new Fundamental Theory (not theorem) of Arithmetics?

It looks to me like revolution isrequired such as we had by Newton bringing accelleration to mechanics, or by Bohr and Schrödinger bringing quantum to mechanics...

Dont we need a discipline that "observes" the system of numbers (N or P) as a scientist would do? I know this is a great tabu to math! But for a moment isnt that we need strong observationout the cage: the view from top on the labyrinth?

--- In primenumbers@yahoogroups.com, Maximilian Hasler <maximilian.hasler@...> wrote:

>

> These are exactly such questions which are not at all answered neither

> by a formula, nor by the (hypothetical and impossible in view of

> infinitude of primes and finiteness of computers and human mind)

> knowledge of ALL primes (which would be way beyond the formula

> producing "one prime after the other").

>

> Maximilian

>

>

> On Fri, Apr 5, 2013 at 11:12 AM, viva8698 <vaseghisam@...> wrote:

> > This is absolutely exciting.

> > You bring it to the point of "structure"

> > What do you see all as the problem of "structure"

> > We where just having a chat with our colleagues from physics.

> > They seem to have an interseting annecdote:

> >

> > Lets assume the Prime Numbers are distributed in a labyrinth with a crayz/chaotic structure and we leave Mr. X there somwhere inside the Labyrinth. He is not allowed at any point (right or left) to make any test/trials (that means e.g. he is not allowed to go first right and proof and get back and go left if first trial wrong). If his formula works he would find the way to all primes (that will lead him to exit). He can only trust his formula (micro).

> >

> > Would he fail?

> >

> > The knowledge of the structure would be a probably more powerful (macro), in that case he would have an overall plan (view from top) and know the whole topology. Is that what you meant by structure?

> >

> > And one of the colleagues came just back with one question: The structure of primes in what?

> >

> >

> > --- In primenumbers@yahoogroups.com, Maximilian Hasler <maximilian.hasler@> wrote:

> >>

> >> > What would be the consequences if Mr. X would have a formula to calculate

> >> from

> >> > a given prime the whole ordered series of the subsequent primes one after

> >> the other?

> >>

> >> Any of the freely or costly available CAS have this function already

> >> implemented, it's called nextprime()

> >>

> >> (and explicit formulas do exist, although those which remain readable are

> >> much less efficient than those which are less "explicit" but rather written

> >> as algorithm).

> >>

> >> I admit that most nextprime() functions use pseudo-primality tests (for

> >> which no counter example is known and which are less likely to yield a

> >> false positive than the probability of an error due to the computer

> >> hardware).

> >> But does this make a difference for this discussion? If so, in which

> >> respect?

> >>

> >>

> >> > 1- Would we need to solve the Riemann hypothesis

> >>

> >> Depends on what you mean by "need".

> >> I don't think that existence of the nextprime() function (or formula)

> >> implies the RH.

> >>

> >> > 2- Would we need to solve all the mountain of conjectures around the

> >> primes?

> >>

> >> Having a formula which yields all primes one after the other

> >> does not yield a response to many conjectures, I think.

> >>

> >>

> >> > Indeed one major point of discussion was to remind:

> >> >

> >> > 3- What where Euler, Riemann, and all the many great mathematicians

> >> seeking for?

> >>

> >> I did not catch what you reminded about this.

> >>

> >>

> >> (Personally I think they were and are seeking better understanding of

> >> several quite distinct mathematical problems, among which might be the

> >> understanding of the structure of the set of prime numbers.

> >> But even knowing all of the latter (even if it was "all at once" and not

> >> "one after the other")

> >> does not mean to understand much about the structure.

> >> That's quite similar with other branches of science:

> >> Even a most complete set of measurements is not equivalent to an

> >> explanation or understanding.)

> >>

> >> Regards,

> >> Maximilian

> >>

> >>

> >> > Look foward.

> >>

> >>

> >> [Non-text portions of this message have been removed]

> >>

> >

> >

>