## Re: [PrimeNumbers] What if Mr. X would have a formula for the prime series?

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• Wouldn t it be enough if given prime pn the formula calculates pn+1? All tests for primality would have to be reviewed. Any algorithm depending on primality
Message 1 of 16 , Apr 3, 2013
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Wouldn' t it be enough if given prime pn the formula calculates pn+1?

All tests for primality would have to be reviewed. Any algorithm depending on primality testing could get a boost in performance from the formula. Cryptography and on line security could be severely affected.

Knowing that no prime is missing in the sequence someone could be tempted to create a new encoding system for numbers in general, based on prime factors: 10 = (1,0,1) 18 = (1,2) 5488 = (4,0,0,3) somehow avoiding the repeating zeros big numbers could be reduced to short sequences.

________________________________
From: viva8698 <vaseghisam@...>
Sent: Tuesday, April 2, 2013 1:34 PM
Subject: [PrimeNumbers] What if Mr. X would have a formula for the prime series?

Recently we had a great discussion during a meeting with colleague 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?

- easily said he gets the 2 and he calcualtes with his formula 3, 5, 7,...

The question appeared to us at the first glance as a joke but soon brought us to a very serious discussion and an explosion of ideas and questions, as well. On the top of many question marks a couple of them were quite hot:

1- Would we need to solve the Riemann hypothesis then at all even if Mr. X and his solution does not provide any way to connect to Zeta?
2- Would we need to solve all the mountain of conjectures around the primes? And if yes which one?
- ...

Indeed one major point of discussion was to remind:

3- What where Euler, Riemann, and all the many great mathematicians seeking for? And what are they still honestly seeking for, when it comes to primes?

Of course there was some bizar questions as well:

- which price would Mr. X be awarded? - Surely not the Nobel price!
- how should Mr. X publish his formula? or should he keep it for himself?

I send out this annecdotical theme and beg all of you so much for your scenarios at least for the questions 1, 2 and 3.

I am sure this can get a great discussion.

Look foward.

[Non-text portions of this message have been removed]
• ... Something like that is used in Fractran (but in reverse) :-) http://en.wikipedia.org/wiki/Fractran yg
Message 2 of 16 , Apr 3, 2013
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Le 2013-04-03 19:40, Leonel Morales a écrit :
> Knowing that no prime is missing in the sequence someone could be
> tempted to create a new encoding system for numbers in general, based
> on prime factors: 10 = (1,0,1) 18 = (1,2) 5488 = (4,0,0,3) somehow
> avoiding the repeating zeros big numbers could be reduced to short
> sequences.

Something like that is used in Fractran (but in reverse) :-)

http://en.wikipedia.org/wiki/Fractran

yg
• Dear Kermit, i could not contact you by your old email adress, neither by kermit@polaris.net nor by the prime forum. Please send me your new mail adress to
Message 3 of 16 , Apr 4, 2013
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Dear Kermit,

neither by kermit@... nor by the prime forum.

pi@...

The best greetings from the primes
Bernhard Helmes
• In reply to What if Mr. X would have a formula for the prime series , please see chapters 2 and 3 of my book Prime Numbers Characteristics -Why They are
Message 4 of 16 , Apr 4, 2013
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In reply to "What if Mr. X would have a formula for the prime series", please see chapters 2 and 3 of my book "Prime Numbers' Characteristics -Why They are what they are".
L.J.Balasundaram

[Non-text portions of this message have been removed]
• ... from ... the other? Any of the freely or costly available CAS have this function already implemented, it s called nextprime() (and explicit formulas do
Message 5 of 16 , Apr 4, 2013
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> 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?

(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]
• 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
Message 6 of 16 , Apr 5, 2013
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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?
>
>
>
> (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]
>
• 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
Message 7 of 16 , Apr 5, 2013
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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?
>>
>>
>>
>> (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]
>>
>
>
• 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
Message 8 of 16 , Apr 5, 2013
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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?
> >>
> >>
> >>
> >> (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]
> >>
> >
> >
>
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