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Polynomial and primes

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  • 逢绥 刘
    It is an old problem in this forum, how much primes a polynomial does represent. Admissible irreducible polynomials with integral coefficients and positive
    Message 1 of 2 , Dec 5, 2006
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      It is an old problem in this forum, how much primes

      a polynomial does represent.



      Admissible irreducible polynomials with integral

      coefficients and positive leading coefficients,

      if it represents a prime, then it represents infinitely many primes.

      This is a party of Schinzel's hypothesis. I am liberty

      to attack it in the paper “There are infinitely many primes of the form a^{2}+1”.

      http://www.paper.edu.cn/privatetools/released/download.jsp?file=200611-738

      Welcome point out any error.

      Original Schinzel's hypothesis: Admissible irreducible polynomials with

      integral coefficients and positive leading coefficients represent infinitely many primes.

      It seem this involved the Ross-Littwood paradox, which is a contradiction between

      numerical limit (not equal 0) and set theoretic limit(empty), like Phil has pointed out.

      I think that it is impossible to prove it.

      Example: we have not proved that there is number a > 41 such that

      x^2 �Cx +a represents primes for x=0,1,2,…,a. If we find such number a or prove there

      is such number a, then there are infinitely many number x such that x^2 �Cx +a represents

      primes for x=0,1,2,…,a.

      I formalized Ross-Littwood paradox in second order formal system P(N), can use

      prime patterns to abstain the paradox.



      If delete condition “irreducible”, then we say that a composite polynomial,

      except the finite primes of the form 1*f_2(x), has not prime patterns of the

      form f_1(x)*f_2(x) for f_1(x)>1 and f_2(x) >1.

      We have a conclusion:

      Admissible polynomials with integral coefficients and positive leading coefficients,

      if it has a prime pattern, then it represents infinitely many primes.

      Welcome anyone continues discussion of this problem.



      I admit my mistake and revise it. Thank Kermit.



      Liu Fengsui.




      ---------------------------------
      Mp3疯狂搜-新歌热歌高速下

      [Non-text portions of this message have been removed]
    • Joshua Zucker
      ... If I understand correctly, then the proof mentioned at http://mathworld.wolfram.com/LuckyNumberofEuler.html shows that there is no number 41 that works
      Message 2 of 2 , Dec 5, 2006
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        On 12/5/06, 逢绥 刘 <liu_f_s@...> wrote:
        > Example: we have not proved that there is number a > 41 such that
        >
        > x^2 �Cx +a represents primes for x=0,1,2,…,a. If we find such number a or prove there
        >
        > is such number a, then there are infinitely many number x such that x^2 �Cx +a represents
        >
        > primes for x=0,1,2,…,a.

        If I understand correctly, then the proof mentioned at
        http://mathworld.wolfram.com/LuckyNumberofEuler.html
        shows that there is no number > 41 that works here (though of course
        you meant not to include a in the list, since a^2 - a + a is surely
        not prime).

        --Joshua Zucker
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