Prove that an odd integer $n>1$ is prime if and only if it is not expressible as a sum of three or more consecutive integers. Prove that an odd integer $n>1$ is prime if and only if it is not expressible as a sum of three or more consecutive integers.
I can see how this works with various examples of the sum of three or more consecutive integers being prime, but I can't seem to prove it for all odd integers $n>1$. Any help would be great.

Hint: The number of terms in the sum and the middle term are especially relevant.
To elaborate a little, $m\cdot n = n + n + \cdots +n = \cdots + (n-1) + n + (n+1) + \cdots$.
Conversely, $(n-k) + \cdots + (n+k) = \cdots$.
