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Conjecture on when consecutive primes are twin primes

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  • Kermit Rose
    RE twin primes conjecture: A property of twin primes, only? Posted by: Jose Ramón Brox ambroxius@terra.es ambroxius Date: Wed Jul 5, 2006 4:51 am (PDT) ...
    Message 1 of 1 , Jul 5, 2006
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      RE twin primes conjecture: A property of twin primes, only?
      Posted by: "Jose Ramón Brox" ambroxius@... ambroxius
      Date: Wed Jul 5, 2006 4:51 am (PDT)



      > Let p q consecutive prime numbers p<q.
      > Let z=sqrt[(p^2+q^2)/2-1]
      >


      > Conjecture: p&q ares twin primes IF AND ONLY IF z is
      > integer.




      No, it's not! You are forgetting the first line of the conjecture. I will
      restate it:

      KNOWING that p and q are consecutive prime numbers with p < q, then p and q
      are twins iff
      z=sqrt(p^2+q^2)/2-1) is integer.

      Your counterexample isn't valid, since it doesn't fulfill the first
      condition: 23 and 25
      they aren't consecutive primes!

      You can look at it in another way: you select a prime number, and scan for
      the next one;
      if you compute z and it's an integer, can you conclude that q-p = 2?

      Regards. Jose Brox


      *************

      Thanks for the clarification.

      I attempted to analyse this.

      Consecutive primes would be of form

      p = 6 D -1 and q = 6 D + 6 m + 1

      or

      p = 6 D -1 and q = 6 D + 6 m + 5.

      Perhaps it may be possible to prove that in both cases that if m > 0, that

      z = sqrt( [ p^2 + q^2]/2 -1 ) is not rational

      and

      also to prove that


      p = 6 D - 1 and q = 6 D + 5

      yields a non-rational z.


      My suggestion consists of trying to prove

      not that z is a non-integer,

      but that z is non-rational.


      If the proof falls through, the attempted proof might suggest how to locate
      the

      counter examples.


      Kermit < kermit@... >
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