What degrees are possible for maps $\mathbb R \mathbb P^{2n+1} \rightarrow \mathbb R \mathbb P^{2n+1}$?
I'm asking about the odd dimensions because we cannot define degree (in a way that would make this question interesting) for the even dimensional real projective spaces, as those manifolds are not orientable (over $\mathbb Z$).
When $n=0$, we get a circle and all degrees are possible.
For larger $n$, I'm not sure how to proceed. We don't get any useful information from the cohomology ring over $\mathbb Z_2$ (and I'm not sure what the ring looks like over $\mathbb Z$). I suspect there are several restrictions on the degree based on the analogous question for $\mathbb C \mathbb P^n$, where only $n$th powers are possible. (To see this, note that any map induces an endomorphism of the cohomology ring $\mathbb Z[x]/(x^{n+1})$ over $\mathbb Z$. One can construct explicit examples showing that all of these degrees can be realized.) However, the present might be easier because the manifolds are not simply connected, so covering space theory may be applied.
This is just an idle question. I wonder more generally about the kinds of tools that exist to answer this and similar questions for different manifolds. The examples of degree classifications I've seen seem pretty contrived.
Take $\mathbb Z_2$-factoring $S^{2n+1}\to\mathbb RP^{2n+1}$. Construct map $S^{2n+1}$ to $S^{2n+1}$ as $2n$-suspension of map $S^1\to S^1$ that has degree $k$.
EDIT: we can correctly define the factor $\mathbb RP^{2n+1}\to\mathbb RP^{2n+1}$. Now show that it has degree $k$. The composition $S^{2n+1}\to S^{2n+1}\to\mathbb RP^{2n+1}$ has degree $2k$, so $S^{2n+1}\to\mathbb RP^{2n+1}\to\mathbb RP^{2n+1}$ has degree $2k$ too.