## Re: [PrimeNumbers] Equivalence for twin primes

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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
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@...>
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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