Roshan's Conjecture

When I was a young child, I noticed a pattern. Every number past 3 seemed to have a pair of primes 'around' it.

1. $(4-1)<4<(4+1)$
2. $(5-2)<5<(5+2)$
3. $(6-1)<6<(6+1)$
4. $(7-4)<7<(7+4)$
5. $(11324-45)<11324<(11324+45)$

Notice how for every number you can think of, there's a prime below it and a prime above it equidistant from it.

Better stated, what I concluded was that:

$\mathrm{\forall }n\in N>3,\mathrm{\exists }k\in N\ge 1$ such that $n-k$ and $n+k$ are both prime. I couldn't find a single counter-example to this. I was terribly excited, of course, because I thought I'd discovered some fundamental property of the universe.

Of course this turns out to be almost entirely equivalent to the famous modern form of Goldbach's conjecture with a little cajoling. So much for my foray into Mathematics!