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Is it known if for every positive integer $k$ there is a positive integer $n$ such that all the numbers $2n,2n+2,\ldots,2n+2k$ are sum of two primes?

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Refer to Goldbach Conjecture http://en.wikipedia.org/wiki/Goldbach's_conjecture . It says that any even number greater than 2 can be represented as sum of $2$ primes. This has been verified for numbers upto $4.10^{17}$ I hope that you understand that the numbers you are asking of are all even. But, no formal proof is available. This is supposed to be true and are supported by some Heuristic Proofs.

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  • $\begingroup$ Of course Goldbach's conjecture is somewhat related to this question, but don't you think it could be proven without Goldbach? With Chen's theorem for example, I'm not sure. $\endgroup$ – punctured dusk Jan 24 '14 at 11:00
  • $\begingroup$ I agree. But, even Chen's Theorem has a limitation,the 'semiprime' part. But, I am not sure too. $\endgroup$ – Hawk Jan 24 '14 at 11:03
  • $\begingroup$ But, one thing is true, if this proof can be produced, there will be a mathematical breakthrough. For example, if $k>4.10^{17}$ $\endgroup$ – Hawk Jan 24 '14 at 11:08
  • $\begingroup$ I know that Goldbach's conjecture will imply my conjecture. I have also found some proved theorems similar to Goldbach's conjecture but I haven't found if the version I have on my mind is open. $\endgroup$ – selfstudying Jan 24 '14 at 11:26
  • $\begingroup$ what do you mean by 'open'? $\endgroup$ – Hawk Jan 24 '14 at 11:29

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