## Equivalence for twin primes

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• 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]
• { 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/
>
>
>
>
>
• ... 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]
• 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.
•  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@...>
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/
>