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Re: Is Goldbach’s conjecture a special case of the twin primes p

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  • Kermit Rose
    Suppose that we identify primes to be both positive and negative integers, p, such that there does not exist non-units j and k, such that j * k = p. There
    Message 1 of 1 , Jun 1, 2013
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      Suppose that we identify primes to be both positive and negative
      integers, p, such that
      there does not exist non-units j and k, such that j * k = p.


      There could be some argument about whether or not to admit (-1) as a prime.

      It is useful in some cases to include (-1) as a prime.

      In either case,

      The twin prime conjecture, "there exist infinitely many prime pairs such
      that their difference is equal to 2",

      becomes indistinguishable from

      "there exist infinitely many prime pairs such that there sum is 2."

      On the other hand,

      Goldbach's conjecture, "For each even number, there exist some pair of
      primes which add to that even number"
      becomes indistinguishable from,

      "For each even number, there exist some pair of primes such that their
      difference is equal to that even number."


      Thus both Goldbach's conjecture and the twin prime conjecture become
      implied by

      "For each even integer there exist infinitely many primes such that
      their sum or difference is equal to that even number".


      Kermit
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