# Does Goldbach conjecture imply the twin prime conjecture ?

Under the Goldbach conjecture, define a primality radius of a large enough composite positive integer $$n$$ to be a positive integer $$r$$ such that both $$n-r$$ and $$n+r$$ are primes. Assume $$k$$ is such that $$1$$ is a primality radius of $$k$$, that is, such that there exists a couple of twin primes whose half sum is $$k$$. Does there, under these assumptions, always exist some integer $$m>k$$ such that $$1$$ , $$k-1$$ and $$k+1$$ are primality radii of $$m$$? If yes, this would imply, under Goldbach conjecture, that there are infinitely many twin primes.

Example :$$k=42$$, half sum of twin primes $$41$$ and $$43$$. One can take $$m=60$$ to get the twin primes $$17$$ and $$19$$ , $$59$$ and $$61$$, $$101$$ and $$103$$.

In case of an affirmative answer to my question, I'd also be interested in an upper bound of the smallest such $$m$$ in terms of $$k$$.