Unique Sequence of Consecutive Odd Numbers which are Prime

Theorem
For $n > 3$ the integers $n$, $n+2$, $n+4$ cannot all be prime. In other words, three consecutive primes do not exist expect for {3,5,7}.

Proof
Assume $n$, $n+2$, and $n+4$ are all prime and n > 3. Any prime number n can be represented as either: $n = 3k$, or $n=3k+1$, or $n=3k+2$.

If $n=3k$, then n is not prime since $3 | 3k$

If $n=3k+1$, then n+2 is not prime since $3|3k+3$

If $n=3k+2$, then n+4 is not a prime since $3|3k+6$

Therefore there doesn't exist any prime number n for which n, n+2, and n+4 are all prime.