Definition Let $M$ be an $n$ dimensional manifold. If $M$ = $M_1$ # $M_2$, we have $M_1=\mathbb S^n$ or $M_2=\mathbb S^n$. Then $M$ is called a prime manifold.
$\mathbb T^2$ and $\mathbb RP^2$ are prime from the classification of surface and $\mathbb S^n$ is prime. So my question is:
Are $\mathbb T^3$ and $\mathbb RP^3$ prime? Also $\mathbb T^n$ and $\mathbb RP^n$?
Is $\mathbb R^n$ prime?
