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A property of twin primes, only?

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  • Sebastian Martin
    Let p q consecutive prime numbers p
    Message 1 of 2 , Jul 4 8:03 AM
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      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.


      This is not a trivial result.

      Sincerely
      Sebastian Martin Ruiz


      See you below:


      The "only if" is trivial:
      If p and q are twin primes, then q = p+2, so z =
      (p+1).

      As for the "if":
      The conjecture is true for p = 2, so assume p >= 3.

      If we write
      q = p+2*d
      then
      z^2 = (p+d)^2 + (d^2-1)

      If (p+d)^2 + (d^2-1) < (p+d+1)^2, then z^2 is not a
      perfect square.
      This condition is:
      q < p + 2*(sqrt(2*p+3)+1)
      which is easily verified to hold for p <= 10^9.
      It is believed (but has never been proved) to hold for
      all p >= 3.

      -Mike Oakes





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    • Sebastian Martin
      I dont spoke about you. There are other e-mails in this list that said this result is trivial. You don t understand me correctly. Sorry my english is very
      Message 2 of 2 , Jul 4 10:14 AM
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        I dont spoke about you. There are other e-mails in
        this list that said this result is trivial.
        You don't understand me correctly. Sorry my english is
        very little.

        Your e-mail is very intersting. I put it by a correct
        example.

        I am sorry that I have been understood badly.
        --- mikeoakes2@... escribió:

        > Sebastian
        >
        > I didn't say it was trivial.
        >
        > I said
        > "p&q are twin primes => z is integer"
        > is trivial.
        >
        > Isn't it??
        >
        > Mike Oakes
        >
        >
        > -----Original Message-----
        > From: Sebastian Martin <sebi_sebi@...>
        > To: lista de primos <primenumbers@yahoogroups.com>
        > Sent: Tue, 4 Jul 2006 17:03:44 +0200 (CEST)
        > Subject: [PrimeNumbers] A property of twin primes,
        > only?
        >
        >
        > 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.
        >
        > This is not a trivial result.
        >
        > Sincerely
        > Sebastian Martin Ruiz
        >
        > See you below:
        >
        > The "only if" is trivial:
        > If p and q are twin primes, then q = p+2, so z =
        > (p+1).
        >
        > As for the "if":
        > The conjecture is true for p = 2, so assume p >= 3.
        >
        > If we write
        > q = p+2*d
        > then
        > z^2 = (p+d)^2 + (d^2-1)
        >
        > If (p+d)^2 + (d^2-1) < (p+d+1)^2, then z^2 is not a
        > perfect square.
        > This condition is:
        > q < p + 2*(sqrt(2*p+3)+1)
        > which is easily verified to hold for p <= 10^9.
        > It is believed (but has never been proved) to hold
        > for
        > all p >= 3.
        >
        > -Mike Oakes
        >
        >
        > ______________________________________________
        > LLama Gratis a cualquier PC del Mundo.
        > Llamadas a fijos y móviles desde 1 céntimo por
        > minuto.
        > http://es.voice.yahoo.com
        >
        >
        >




        ______________________________________________
        LLama Gratis a cualquier PC del Mundo.
        Llamadas a fijos y móviles desde 1 céntimo por minuto.
        http://es.voice.yahoo.com
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