Unique Sequence of Consecutive Odd Numbers which are Prime

Theorem
Let $n \in \Z$ be an integer such that $n > 3$.

Then $n$, $n+2$, $n+4$ cannot all be prime.

That is, three consecutive odd integers cannot all be prime except for $\left\{{3, 5, 7}\right\}$.

Proof
Let $n \in \Z_{>0}$.

If $n \le 3$ the the theorem clearly holds.

Suppose $n > 3$.

Any integer $n$ can be represented as either:

If $n=3k$, then $n$ is not prime since $3 \mathop \backslash 3k$.

If $n=3k+1$, then $n+2$ is not prime since $3 \mathop \backslash 3k + 3$.

If $n=3k+2$, then $n+4$ is not a prime since $3 \mathop \backslash 3k + 6$.

Therefore no such $n$ exists for which $n$, $n+2$, and $n+4$ are all prime.