Is it true that every compact and orientable $n$-manifold has a prime decomposition (a decomposition as a connected sum of prime manifolds)?
Here, an $n$-dimensional manifold $M$ is called prime (in a suitable category: topological manifolds, PL manifolds, smooth manifolds) if whenever $M= M_1\# M_2$, one of the manifolds $M_i$ is the $n$-sphere. The existence of a prime decomposition in dimensions $\le 3$ is well-known (and even has its own Wikipedia page).
Assuming that you are working with topological manifolds, the answer is positive. See Corollary 2.5 in
I. Bokor et al, Connected sum decompositions of high-dimensional manifolds.
The key ingredients to the proof are Grushko's theorem, Hurewicz theorem and topological Poincare conjecture.
However, if you are thinking about differentiable manifolds then the answer is again positive in all dimensions $\ne 4$ and is very difficult (smooth Schonflies conjecture) in dimension 4.