If the set of primes where $p$, $p+2$ is infinite, would this imply that the set of $p$ and $p+2n$ is also infinite?

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?

• Is $n$ a fixed positive integer? Jan 22, 2013 at 7:01
• do you mean "the set of numbers $p$ such that $p$ and $p+2$ are prime is infinite"? Jan 22, 2013 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. Jan 22, 2013 at 7:05
• Also, nothing of this type has been proved; see en.wikipedia.org/wiki/Twin_prime Jan 22, 2013 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. Jan 22, 2013 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.

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.

• I think it is safer to bet on $2^n$ instead of $2n$. Based on Hardy-Littlewood.
– mick
Sep 4, 2013 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.

• 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.
– user14972
Jan 22, 2013 at 12:02
• Yes, but the question asks whether the result itself implies the more general result. Jan 22, 2013 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? Jan 22, 2013 at 12:04
• Oh I see the point now, I was getting confused with the use of $p$ twice in the question. Jan 22, 2013 at 12:41