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Re: [PrimeNumbers] Equivalence for twin primes

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  • 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 1 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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