# How many consecutive even numbers are sum of two primes?

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?

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.
• But, one thing is true, if this proof can be produced, there will be a mathematical breakthrough. For example, if $k>4.10^{17}$ – Hawk Jan 24 '14 at 11:08