In how many ways can $10001$ be written as the sum of two primes?
Obviously since the 10001 is odd, one of the primes must be $2$. This leaves the second, must be prime as 9999, but it isn't, hence there are $0$ ways to write $10001$ as the sum of two primes.
Another way I would appreciate if someone can give me feedback on is:
All primes $p$ can be expressed as either $p=6k+1$ or $p=6k-1$, where $k$ is some positive integer. If I sum two distinct primes I have three possibilities: $$p_1+p_2=(6k+1)+(6m+1)=6(k+m)+2$$ or $$p_1+p_2=(6k-1)+(6m+1)=6(k+m)$$ or $$p_1+p_2=(6k-1)+(6m-1)=6(k+m)-2$$
Since $10001$ is neither even, which would satifie (1) or (3) and since it is not multiple of 6, which would satisfy (2), there exists no primes that would sum to $10001$. Is this logic correct?