John, it's only the counting problem that would provide a way to factor RSA numbers, and the efficient algorithms don't provide any way to get a count.

The two answers skip over $p=2$ and $p=3$, which are the most interesting answers (though Wikipedia describes them in part)

Not sure about a citable reference but in general for non-orientable manifolds there always exists an orientable double cover.

The standard infinite product for the cosine is, IIRC, $\cos(\pi x)=\prod_{n=0}^\infty 1-\frac{(2x)^2}{(2n+1)^2}$

A probabilistic resolution would be to, say, assume the phase takes a value based on a circular Cauchy distribution; then you just have to take a map from the Moebius band to the disc, which is possible.

3x3 is the minimum you need to see an "edge" of bright-dark-bright or dark-bright-dark which is one of the axes of the Klein bottle.

I think the term you want to look up is "systolic".

I could get useful information out of finite fields, probably. How much faster is that usually? Also the rings I'm working with have lots of symmetries and I think the letterplace correspondence gives even more in the actual ring being computed. How do you factor those out?

I think triality more often than not refers to operations involving the exceptional degree-$3$ outer automorphism of $Spin(8)$.

Yeah, the group algebra of $Z/2$ is basically your only option in characteristic different from $2$. In characteristic $2$ you have a couple options though.

It sounds like the Super Monster just became the Monster and it was the Middle Monster that disappeared...

I'm still waking up but I think such algebras are always both commutative and cocommutative, since the identity now has to be an anti-isomorphism, severely limiting your options.

Consider $\mathbf{C}^2 \setminus 0$, which is topologically a $3$-sphere.

Heck, Quaternion-Kaehler manifolds are scarce: I don't think we know of any other than the Wolf manifolds...