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

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  • Leonel Morales
    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@...>
      To: primenumbers@yahoogroups.com
      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]
    • yasep16
      ... 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
      • bhelmes_1
        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,

          i could not contact you by your old email adress,
          neither by kermit@... nor by the prime forum.

          Please send me your new mail adress to
          pi@...

          The best greetings from the primes
          Bernhard Helmes
        • LALGUDI BALASUNDARAM
          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]
          • Maximilian Hasler
            ... 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?

              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]
            • viva8698
              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?
                >
                > 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]
                >
              • Maximilian Hasler
                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?
                  >>
                  >> 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]
                  >>
                  >
                  >
                • viva8698
                  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?
                    > >>
                    > >> 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]
                    > >>
                    > >
                    > >
                    >
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