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Re: Goldbach Proof

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  • Mike Oakes
    ... Conjecture ... [I typed a reply on the website about 2 hours ago but must have hit the wrong key as it seems to have disappeared into the ether without
    Message 1 of 4 , Mar 3, 2007
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      --- In primenumbers@yahoogroups.com, "Werner D. Sand"
      <Theo.3.1415@...> wrote:
      >
      > What do you think about the so-called proof of the Goldbach
      Conjecture
      > by Jinzhu and Zaizhu Han?
      > http://arxiv.org/ftp/math/papers/0701/0701235.pdf
      >

      [I typed a reply on the website about 2 hours ago but must have hit
      the wrong key as it seems to have disappeared into the ether without
      trace.
      Apolgies if it also turns up some time.] I wrote something like this:

      Their whole claim to a proof rests on the statement made after their
      eqn. (1.6):
      if pi(N,lambda) > 0 (for all N), then GC is true.

      I think this is claimed implication is false.

      Take a particular example: N = 10.

      Then their definition of lambda(n), eqn. (1.1), says that
      lambda(n) = 0 if n = 1 mod 3, else 1.

      By (1.3),
      pi(N,lambda) = sum_{p <= N} lambda(p)
      = lambda(3) + lambda(5) + lambda(7)
      = 1 + 1 + 0.

      This is indeed > 0 but so what?
      That's NOT the same as the fact that there are primes p1 and p2 < 10
      such that
      p1 + p2 = 10.

      So, proving that pi(N,lamda) > 0 for all (sufficiently large) N,
      which they MAY have done (giving the benefit of the doubt to their
      rather difficult proof steps), is NOT equivalent to proving Goldbach.

      Moreover, GC is for /all/ N, not just for large N, and since their
      proof steps only apply for large N they fall down on that score, at
      the very least.

      -Mike Oakes
    • Jens Kruse Andersen
      ... After briefly looking at later parts, I guess misdefined. When defining lambda(n), they probably assume N is an already given constant (the number they
      Message 2 of 4 , Mar 3, 2007
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        Phil Carmody wrote:
        > --- "Werner D. Sand" <Theo.3.1415@...> wrote:
        > > What do you think about the so-called proof of the Goldbach
        > > Conjecture by Jinzhu and Zaizhu Han?
        > > http://arxiv.org/ftp/math/papers/0701/0701235.pdf
        >
        > Well, they aren't loons as they are familiar with prior work in
        > the area; but I can't help thinking that a unary function which
        > is defined in terms of two unknowns is going to lead to problems:
        >
        > lambda(n) = { 0, if n==N mod p, p<=sqrt(N), (N,p)=1
        > { 1, otherwise
        >
        > So, chose p such that (n,p)=1, and then set N=p^2+(n%p)
        > Then p<=sqrt(N), and (N,p)=(n,p)=1
        >
        > So their lambda(n) is either trivially 0 everywhere, or misdefined.

        After briefly looking at later parts, I guess misdefined.
        When defining lambda(n), they probably assume N is an already given
        constant (the number they want to write as sum of two primes), while
        p can be any prime.
        Under this assumption, if n < N and lambda(n) = 1, then N-n is prime.
        I will not review the paper.

        --
        Jens Kruse Andersen
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