How to prove the following conjecture:
For any positive integer $n>3$, there exists at least $1$ integer $k$ such that $n+k$ and $n-k$ are both primes.
Any hint, idea or reference would be greatly appreciated!
I suspect the OP means $n + k$ AND $n - k$. If so, this is the same as the strong Goldbach conjecture, which is a very well-known unsolved problem. In particular, if $n + k$ and $n - k$ are both prime, their sum is $2n$. Also, every positive even integer $\gt 2$ is of the form $2n$, for some positive integer $n$. Thus, if the OP's conjecture holds, so does the Goldbach conjecture.
I would do some searching in the On-Line Encyclopedia of Integer Sequences (OEIS). Give me a couple of minutes...
Okay, here's one pertinent result: http://oeis.org/A020483 Least prime $p$ such that $p + 2n$ is also prime. According to Jens Kruse Andersen, "It is conjectured that $a(n)$ always exists."