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Equivalence for twin primes

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  • Sebastian Martin Ruiz
    Hello all: Let n and k positive integers k
    Message 1 of 5 , Mar 5, 2011
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      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]
    • Maximilian Hasler
      { 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] ,
      Message 2 of 5 , Mar 5, 2011
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        { 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]
        >
        >
        >
        > ------------------------------------
        >
        > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
        > The Prime Pages : http://www.primepages.org/
        >
        > Yahoo! Groups Links
        >
        >
        >
        >
      • 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 3 of 5 , Mar 5, 2011
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          > { 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 4 of 5 , Mar 5, 2011
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            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 5 of 5 , Mar 5, 2011
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               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]
              >
              >
              >
              > ------------------------------------
              >
              > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
              > The Prime Pages : http://www.primepages.org/
              >
              > Yahoo! Groups Links
              >
              >
              >
              >






              [Non-text portions of this message have been removed]
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