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Re: [PrimeNumbers] Equivalence for twin primes

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  • Peter Kosinar
    ... I believe the Sebastian meant ... IFF P(n) and P(k) form a Twin pair , i.e. P(n) - P(k) = 2. The IF direction is trivial, of course. Peter [Non-text
    Message 1 of 5 , Mar 5, 2011
      > { for(n=1,9,Pn=prime(n);for(k=1,n-1,Pk=prime(k);Pnk=prime(n-k);
      > (Pn-Pnk==(n-k)*Pk) == (istwin(Pn)&istwin(Pk)) |
      > print("Counter-example: (n,k)=",[n,k]",
      > P(n)-P(n-k)-(n-k)*P(k)=",Pn"-"Pnk"-",n-k,"*"Pk))) }
      >
      > [lots of counterexamples]

      > > P(n)-P(n-k)-(n-k)P(k)=0   IF AND ONLY IF    P(n) and P(k) are Twin Primes

      I believe the Sebastian meant "... IFF P(n) and P(k) form a Twin pair",
      i.e. P(n) - P(k) = 2. The "IF" direction is trivial, of course.

      Peter

      [Non-text portions of this message have been removed]
    • Kermit Rose
      On 3/5/2011 8:04 AM, Sebastian Martin Ruiz s_m_ruiz@yahoo.es s_m_ruiz ... It is trivial that if p(n) , p(n-k), and p(k) are distinct primes, and n k, then
      Message 2 of 5 , Mar 5, 2011
        On 3/5/2011 8:04 AM, "Sebastian Martin Ruiz" s_m_ruiz@... s_m_ruiz
        wrote:
        > ________________________________________________________________________
        > 1. Equivalence for twin primes
        > Posted by: "Sebastian Martin Ruiz" s_m_ruiz@... s_m_ruiz
        > Date: Sat Mar 5, 2011 2:13 am ((PST))
        >
        > Hello all:
        >
        > Let n and k positive integers k<n.
        >
        > Let P(i) the ith-prime number
        >
        > We have:
        >
        > P(n)-P(n-k)-(n-k)P(k)=0 IF AND ONLY IF P(n) and P(k) are Twin Primes
        >
        >
        > Sincerely
        >
        > Sebastián Martín Ruiz
        >
        >
        >


        It is trivial that

        if p(n) , p(n-k), and p(k) are distinct primes,
        and n>k,
        then
        P(n)-p(n-k) -(n-k)P(k)=0

        if and only if p(n-k) = 2, and n-k = 1.




        In the case where p(n) and p(k) belong to the same set of twin primes,
        we would have

        example:
        p(n) = 7, p(k) = 5

        n = 4, k = 3

        p(n-k) = p(1) = 2

        P(n)-2 -(n-k)P(k)=0

        7 - 2 - 1*5 = 5 - 5 = 0

        In general , if p(n) and p(k) belong to the same set of twin primes,
        and n > k,

        it is trivial that

        P(n)-P(n-k)-(n-k)P(k)
        = p(n) - p( n - [n-1]) - ([n-(n-1)] )p(n-1)
        = p(n) - p(1) - 1 * (p(n-1))
        = p(n) - 2 - p(n-1)
        = (p(n) - p(n-1)) - 2
        = 2 - 2 = 0


        If
        P(n)-P(n-k)-(n-k)P(k) = 0
        p(n) - p(n-k) = (n-k) p(k)

        n = 2
        what are permitted values of k?
        k = 1?
        3 - p(1) = (2-1)*p(1) ?
        3 - 2 = 1 * 2 ?
        1 = 2 ?
        no

        n = 3
        p(3) - p(3-k) - (n-k) p(k) = 0
        p(3) - p(3-k) = (n-k) p(k)
        5 - p(3-k) = (n-k)

        5 = p(3-k) + (3-k)
        2 = p(3-k) - k

        2 + k = p(3-k)

        k is odd



        In the case where p(n) and p(k) belong to different
        sets of twin primes,

        example:

        p(n) = 13
        p(k) = 7

        n = 6
        k = 4
        n-k = 2

        P(n)-P(n-k)-(n-k)P(k)
        =13 - 3 - 2*7
        = -4
        is not zero.
      • Sebastian Martin Ruiz
         P(n)-P(n-k)-(n-k)P(k)=0   IIF    P(n) and P(k) are a Twin Primes pair IF P(n) and P(k) are a Twin Primes pair Then k=n-1 P(n-k)=P(1)=2 
        Message 3 of 5 , Mar 5, 2011
           P(n)-P(n-k)-(n-k)P(k)=0   IIF    P(n) and P(k) are a Twin Primes pair

          IF P(n) and P(k) are a Twin Primes pair

          Then k=n-1 P(n-k)=P(1)=2  P(n)-2-(n-(n-1))P(n-1)=0

          Then P(n)-P(n-1)=2

          On the other hand If

          P(n)-P(n-k)-(n-k)P(k)=0  then

           P(n) and P(k) are a Twin Primes pair

           is more dificult but i think it is also true.


          P(n)-P(n-k)-(n-k)*P(k)=7-3-2*3=/=0
          Counter-example: (n,k)=[4, 2], P(n)-P(n-k)-(n-k)*P(k)=7-3-2*3
          then your contraexample is not true.





           



          ________________________________
          De: Maximilian Hasler <maximilian.hasler@...>
          Para: Sebastian Martin Ruiz <s_m_ruiz@...>
          CC: primenumbers@yahoogroups.com
          Enviado: sáb,5 marzo, 2011 14:12
          Asunto: Re: [PrimeNumbers] Equivalence for twin primes

           
          { for(n=1,9,Pn=prime(n);for(k=1,n-1,Pk=prime(k);Pnk=prime(n-k);
          (Pn-Pnk==(n-k)*Pk) == (istwin(Pn)&istwin(Pk)) |
          print("Counter-example: (n,k)=",[n,k]",
          P(n)-P(n-k)-(n-k)*P(k)=",Pn"-"Pnk"-",n-k,"*"Pk))) }

          Counter-example: (n,k)=[4, 2], P(n)-P(n-k)-(n-k)*P(k)=7-3-2*3
          Counter-example: (n,k)=[5, 2], P(n)-P(n-k)-(n-k)*P(k)=11-5-3*3
          Counter-example: (n,k)=[5, 3], P(n)-P(n-k)-(n-k)*P(k)=11-3-2*5
          Counter-example: (n,k)=[5, 4], P(n)-P(n-k)-(n-k)*P(k)=11-2-1*7
          Counter-example: (n,k)=[6, 2], P(n)-P(n-k)-(n-k)*P(k)=13-7-4*3
          Counter-example: (n,k)=[6, 3], P(n)-P(n-k)-(n-k)*P(k)=13-5-3*5
          Counter-example: (n,k)=[6, 4], P(n)-P(n-k)-(n-k)*P(k)=13-3-2*7
          Counter-example: (n,k)=[7, 2], P(n)-P(n-k)-(n-k)*P(k)=17-11-5*3
          Counter-example: (n,k)=[7, 3], P(n)-P(n-k)-(n-k)*P(k)=17-7-4*5
          Counter-example: (n,k)=[7, 4], P(n)-P(n-k)-(n-k)*P(k)=17-5-3*7
          Counter-example: (n,k)=[7, 5], P(n)-P(n-k)-(n-k)*P(k)=17-3-2*11
          Counter-example: (n,k)=[7, 6], P(n)-P(n-k)-(n-k)*P(k)=17-2-1*13
          Counter-example: (n,k)=[8, 2], P(n)-P(n-k)-(n-k)*P(k)=19-13-6*3
          Counter-example: (n,k)=[8, 3], P(n)-P(n-k)-(n-k)*P(k)=19-11-5*5
          Counter-example: (n,k)=[8, 4], P(n)-P(n-k)-(n-k)*P(k)=19-7-4*7
          Counter-example: (n,k)=[8, 5], P(n)-P(n-k)-(n-k)*P(k)=19-5-3*11
          Counter-example: (n,k)=[8, 6], P(n)-P(n-k)-(n-k)*P(k)=19-3-2*13

          On Sat, Mar 5, 2011 at 6:13 AM, Sebastian Martin Ruiz <s_m_ruiz@...> wrote:
          > Hello all:
          >
          > Let n and k positive integers k<n.
          >
          > Let P(i) the ith-prime number
          >
          > We have:
          >
          > P(n)-P(n-k)-(n-k)P(k)=0   IF AND ONLY IF    P(n) and P(k) are Twin Primes
          >
          >
          > Sincerely
          >
          > Sebastián Martín Ruiz
          >
          >
          >
          >
          > [Non-text portions of this message have been removed]
          >
          >
          >
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          [Non-text portions of this message have been removed]
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