You should really view the $\sigma_i$ as the embeddings of $K$ into $\mathbb{R}$ and not as elements of the Galois group.

Thanks Will. Chevalley's theorem shows that the number of affine solutions is divisible by $p$. I don't see how to use this to deduce that the number of affine solutions is positive!

You're right Felipe! I have to rethink what I want.

The motivation for theses is problems is that they were beyond what known methods for Diophantine equations were capable of at the time. This is still true of the last problem above: $(x^3-1)/(y^3-1)=z^2$. This problem asks about integral points on a surface. Our understanding of the arithmetic of surfaces is still embryonic.

The arithmetic progressions of the form $a^2$, $b^2$, $c^2$, $d^5$ where $a$, $b$, $c$, $d$ are integers with $\gcd(a,b)=1$ are determined here: arxiv.org/abs/0912.2670

The equations $x^2-x=y^5-y$ and ${x\choose2}={y\choose5}$ are solved here: arxiv.org/abs/0801.4459