## primes in arithmetic sequences

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• Hello: Let p and q odd prime numbers p q by Dirichlet theorem exist t1 and t2 positives integers that  p+t1(p-q) and q+t2(p-q) are primes.Can someone prove
Message 1 of 6 , Feb 26, 2009
Hello:
Let p and q odd prime numbers p>q by Dirichlet theorem exist t1 and t2 positives integers that  p+t1(p-q) and q+t2(p-q) are primes.Can someone prove that they exist by the same t?It is to say exists a positive integer t that p+t(p-q) and q+t(p-q) are both primes?
Sincerely
Sebastian Martin Ruiz

[Non-text portions of this message have been removed]
• Wouldn t this imply the twin prime conjecture ? (for p=q+2 it would imply existence of another twin prime pair at (q,p)+2t, and then so on) Regards, Maximilian
Message 2 of 6 , Feb 26, 2009
Wouldn't this imply the twin prime conjecture ?
(for p=q+2 it would imply existence of another twin prime pair at
(q,p)+2t, and then so on)
Regards,
Maximilian

--- In primenumbers@yahoogroups.com, Sebastian Martin Ruiz
<s_m_ruiz@...> wrote:
>
> Hello:
> Let p and q odd prime numbers p>q by Dirichletï¿½theorem exist t1 and
t2 positives integersï¿½that ï¿½p+t1(p-q) and q+t2(p-q) are primes.Can
someone prove that they exist by the same t?It is to sayï¿½exists a
positive integer t that p+t(p-q) and q+t(p-q) are both primes?
> Sincerely
> Sebastian Martin Ruiz
• Twin prime conjecture is precisely what I am trying to prove with this conjecture. We can prove by Dirichlet Theorem that exist the same t?We know by
Message 3 of 6 , Feb 26, 2009
Twin prime conjecture is precisely what I am trying to prove with this conjecture. We can prove by Dirichlet Theorem that exist the same t?We know by  Dirichlet Theorem that exists infinity many t1 y t2 that p+t1(p-q) and q+t2(p-q) are both primes. But there are no two be equal?

--- El vie, 27/2/09, michael_b_porter <michael.porter@...> escribió:

De: michael_b_porter <michael.porter@...>
Asunto: Re: primes in arithmetic sequences
Para: "Sebastian Martin Ruiz" <s_m_ruiz@...>
Fecha: viernes, 27 febrero, 2009 6:02

--- In primenumbers@yahoogroups.com, Sebastian Martin Ruiz
<s_m_ruiz@...> wrote:
> It is to say exists a positive integer t that p+t(p-q) and q+t(p-q)
are both primes?

Suppose that this conjecture is true.  Let (s,s+2) be a pair of twin
primes.  Then by the conjecture (with p=s+2, q=s), there is a positive
integer t such that s+2+2t and s+2t are both prime.  So for each pair
of twin primes, there is a greater pair of twin primes.

So the twin prime conjecture follows from your conjecture.

- Michael Porter

[Non-text portions of this message have been removed]
• Dear Sebastian:     In a recent work J.M. Deshouillers and F. Luca [On the distribution of some means concerning the densitiy, Funct. Approx. Comment. Math.
Message 4 of 6 , Feb 27, 2009
Dear Sebastian:

In a recent work J.M. Deshouillers and F. Luca [On the distribution of some means concerning the densitiy, Funct. Approx. Comment. Math. Volume 39, Number 2 (2008), 335-344.] consider certain means of the values of the Euler function to prove that they are dense modulo one. At the Czech-Slovak Number Theory Conference in August 2007, F. Luca raised the question whether certain other sequences of mean values of the Euler function are uniformly distributed modulo one. Among these are the sequences of arithmetic and geometric means. Recently, J.M. Deshouillers and H. Iwaniec gave a method leading to an affirmative answer for Luca's question in the case of arithmetic mean, and a conditional answer for the case of geometric mean. The aim is to be framiliar with this method. it might be useful on your approach to prove twin prime number conjecture. meanwhile you can keep up yourself in connection with Professor Iwaniec.

Sincerely Yours.

Saeed Ranjbar

[Non-text portions of this message have been removed]
• 2. primes in arithmetic sequences Posted by: Sebastian Martin Ruiz s_m_ruiz@yahoo.es s_m_ruiz Date: Thu Feb 26, 2009 11:15 am ((PST)) Let p and q odd prime
Message 5 of 6 , Feb 27, 2009
2. primes in arithmetic sequences
Posted by: "Sebastian Martin Ruiz" s_m_ruiz@... s_m_ruiz
Date: Thu Feb 26, 2009 11:15 am ((PST))

Let p and q odd prime numbers p>q by Dirichlet theorem exist t1 and t2
positives integers that p+t1(p-q) and q+t2(p-q) are primes.Can someone
prove that they exist by the same t?It is to say exists a positive
integer t that p+t(p-q) and q+t(p-q) are both primes?
Sincerely
Sebastian Martin Ruiz

r1 = p+t1(p-q)
r2 = q+t2(p-q)

r1 - r2 = ( p+t1(p-q) ) - ( q+t2(p-q) )

r1 - r2 = (p - q) + t1 (p-q) - t2(p-q)

r1 - r2 = (1 + t1 - t2 ) (p - q)

If t1 = t2,

r1 - r2 = p - q

The existence of r1 and r2 is implied by the conjecture that
every even integer is the difference of two primes.

Kermit
• Yes, there are common values for t1 y t2, but no-one can prove it yet. I don t expect that this more general statement could be proved prior to proving the
Message 6 of 6 , Feb 27, 2009
Yes, there are common values for t1 y t2, but no-one can prove it yet.
I don't expect that this more general statement could be proved prior
to proving the twin prime conjecture (which is the special case p-q=2
where the values of t1, t2 are most dense - in fact you have a t1 and
a t2 for each prime larger than p resp. q).
Maximilian

On Fri, Feb 27, 2009 at 3:58 AM, Sebastian Martin Ruiz
<s_m_ruiz@...> wrote:
>
> Twin prime conjecture is precisely what I am trying to prove with this conjecture. We can prove by Dirichlet Theorem that exist the same t?We know by  Dirichlet Theorem that exists infinity many t1 y t2 that p+t1(p-q) and q+t2(p-q) are both primes. But there are no two be equal?
>
> --- El vie, 27/2/09, michael_b_porter <michael.porter@...> escribió:
>
> De: michael_b_porter <michael.porter@...>
> Asunto: Re: primes in arithmetic sequences
> Para: "Sebastian Martin Ruiz" <s_m_ruiz@...>
> Fecha: viernes, 27 febrero, 2009 6:02
>
> --- In primenumbers@yahoogroups.com, Sebastian Martin Ruiz
> <s_m_ruiz@...> wrote:
>> It is to say exists a positive integer t that p+t(p-q) and q+t(p-q)
> are both primes?
>
> Suppose that this conjecture is true.  Let (s,s+2) be a pair of twin
> primes.  Then by the conjecture (with p=s+2, q=s), there is a positive
> integer t such that s+2+2t and s+2t are both prime.  So for each pair
> of twin primes, there is a greater pair of twin primes.
>
> So the twin prime conjecture follows from your conjecture.
>
> - Michael Porter
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