Prime 3-manifold

Definition
A compact connected orientable manifold $$M$$ of dimension 3 is termed a prime 3-manifold if the only possible way of expressing $$M$$ as a connected sum of manifolds, both of dimension 3, is as $$M \# S^3$$. In other words, one of the summands of the connected sum must be homeomorphic to $$M$$ and the other summand must be homeomorphic to the 3-sphere $$S^3$$.