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If the set of primes $p$ such that $p+2$ is also prime is infinite, would this imply that the set of primes such that $p+2n$ where $n$ is any positive integer for each pair is also infinite?

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Is $n$ a fixed positive integer? – Calvin Lin Jan 22 '13 at 7:01
do you mean "the set of numbers $p$ such that $p$ and $p+2$ are prime is infinite"? – Trevor Wilson Jan 22 '13 at 7:02
I do not think there is any known relationship between the question of whether there are infinitely many pairs $(n,n+2)$ of primes and the question of whether there are infinitely many pairs $(n,n+4)$ of primes. – André Nicolas Jan 22 '13 at 7:05
Also, nothing of this type has been proved; see – Will Jagy Jan 22 '13 at 7:07
@AndréNicolas, My question is if there are infinitely many primes with difference $2$, Is there a solid relation between that conjecture and the conjecture that there are infinitely many with difference $4,6,8,10,...$? And the same if there are finite. I'm having trouble wording the question and would really appreciate it if you edited the question if you understood me, thanks. – Babiker Jan 22 '13 at 7:27

This could be. It could be that a proof that there are infinitely many primes p and p+2 would imply the proof that there are infinetely many primes p and p+2n for all n = 1,2,3,4,... This is also called sometimes Polignac conjecture.

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So far as I know, no one has ever proved anything along the lines of, "If there are infinitely many pairs of primes differing by $2$, then there are infinitely many pairs of primes differing by $4$."

On the other hand, I don't see what's so special about $2$ (in this context), and I bet that if the day comes when someone produces a proof for $2$, the techniques of that proof will also work for $2n$ generally.

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I think it is safer to bet on $2^n$ instead of $2n$. Based on Hardy-Littlewood. – mick Sep 4 '13 at 20:22

I might be being stupid but surely no large enough twin prime pair (p,p+2) gives a prime pair of difference $4$.

So even if there were infinitely many twin primes, this would tell us nothing about the quantity of difference $4$ primes.

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A priori, anyways. In principle, a proof of the twin prime conjecture could imply results that would also tell us things about the more general case. – Hurkyl Jan 22 '13 at 12:02
Yes, but the question asks whether the result itself implies the more general result. – fretty Jan 22 '13 at 12:04
I think you're missing the point, fretty. True, if $p$ and $p+2$ are both prime, then $p+4$ isn't --- but maybe $2p+1$ and $2p+5$ are both prime. Maybe there is a proof that for .0001 percent of the primes $p$ for which $p+2$ is prime, $2p+1$ and $2p+5$ are both prime. I'm sure no such proof exists now --- are you sure no such proof ever will exist? – Gerry Myerson Jan 22 '13 at 12:04
Oh I see the point now, I was getting confused with the use of $p$ twice in the question. – fretty Jan 22 '13 at 12:41

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