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A different approach to the twin primes conjecture

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  • WarrenS
    Here s a rather different approach than Yitang Zhang s. Probably unlike his it is hopeless, but anyhow I will explain the idea. The goal is to fix a small
    Message 1 of 2 , Jun 4, 2013
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      Here's a rather different approach than Yitang Zhang's.
      Probably unlike his it is hopeless, but anyhow I will explain the idea.

      The goal is to fix a small constant integer K (optimally K=2, but anything
      smaller than, say, 60 will be impressive) then prove that there exist an infinite number of
      prime-pairs {P, P+K}. Call these "K-twins." We also with the approach below could
      be able to attack constellations involving more than two primes, e.g. "triplets"
      and also could attack things like "primes p where 2*p+1 is prime also."

      We shall use theorems about primeness and quadratic forms. Examples include:

      If N=1 mod 4 is squarefree, then N is prime if N has exactly 4 representations
      of the form N=4*a^2+b^2 with a,b integers. Also, it is not possible for
      any such N, whether prime or not, to have 1,2,3,5,6,7 reps.

      If N=1 mod 6 is squarefree, then N is prime if N has exactly 4 representations
      of the form N=3*a^2+b^2 with a,b integers. Also, it is not possible for
      any such N, whether prime or not, to have 1,2,3,5,6,7 reps.

      There are many theorems of this general ilk that arise from unique factorization
      theorems in quadratic number fields with class number 1.

      Now certain "theta functions" give generating functions for these counts of representations. That is of interest because usually with primes you mess with
      zeta functions and Dirichlet L functions, which are horrible. Theta functions
      are far nicer in many ways.

      For example if theta(x)=1+2x+2x^4+2x^9+2x^16+2x^25+...
      then F(x)=theta(x)*theta(x^4) counts representations of N of form N=4*a^2+b^2.

      A difficulty in all this is the fact those theorems only work for SQUAREFREE N.
      However, if we were to prove theorems not about "primes" but rather about "primes and numbers with square factors", ("primeanwsfs") those would still be interesting theorems,
      and you might be able to surmount the difficulty with some kind of sieving argument later (concerning sieving out numbers with square factors).

      Now if F(x) is such a generating function and we want to prove there exist an
      infinite number of primeanwsfs-pairs (P, P+K),
      then one way would be to show an infinite number of powers x^p of x exist in the
      series expansion of (1/2+x^K)*F(x), which have coefficient=6.
    • WarrenS
      Here s a rather different approach than Yitang Zhang s. Probably unlike his it is hopeless, but anyhow I will explain the idea. The goal is to fix a small
      Message 2 of 2 , Jun 4, 2013
      • 0 Attachment
        Here's a rather different approach than Yitang Zhang's.
        Probably unlike his it is hopeless, but anyhow I will explain the idea.

        The goal is to fix a small constant integer K (optimally K=2, but anything
        smaller than, say, 60 will be impressive) then prove that there exist an infinite number of
        prime-pairs {P, P+K}. Call these "K-twins." We also with the approach below could
        be able to attack constellations involving more than two primes, e.g. "triplets"
        and also could attack things like "primes p where 2*p+1 is prime also."

        We shall use theorems about primeness and quadratic forms. Examples include:

        If N=1 mod 4 is squarefree, then N is prime if N has exactly 4 representations
        of the form N=4*a^2+b^2 with a,b integers. Also, it is not possible for
        any such N, whether prime or not, to have 1,2,3,5,6,7 reps.

        If N=1 mod 6 is squarefree, then N is prime if N has exactly 4 representations
        of the form N=3*a^2+b^2 with a,b integers. Also, it is not possible for
        any such N, whether prime or not, to have 1,2,3,5,6,7 reps.

        There are many theorems of this general ilk that arise from unique factorization
        theorems in quadratic number fields with class number 1.

        Now certain "theta functions" give generating functions for these counts of representations. That is of interest because usually with primes you mess with
        zeta functions and Dirichlet L functions, which are horrible. Theta functions
        are far nicer in many ways.

        For example if theta(x)=1+2x+2x^4+2x^9+2x^16+2x^25+...
        then F(x)=theta(x)*theta(x^4) counts representations of N of form N=4*a^2+b^2.

        A difficulty in all this is the fact those theorems only work for SQUAREFREE N.
        However, if we were to prove theorems not about "primes" but rather about "primes and numbers with square factors", ("primeanwsfs") those would still be interesting theorems,
        and you might be able to surmount the difficulty with some kind of sieving argument later (concerning sieving out numbers with square factors).

        Now if F(x) is such a generating function and we want to prove there exist an
        infinite number of primeanwsfs-pairs (P, P+K),
        then one way would be to show an infinite number of powers x^p of x exist in the
        series expansion of (1/2+x^K)*F(x), which have coefficient=6.
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