Martin, I certainly don't insist on putting cohomology into any particular box. My point was that nowadays, people investigate cohomological questions just because they find them interesting, without any topological/geometric/number theoretic applications in the back of their mind and they don't have to justify this "indulgence".

Franz, wasn't Kummer the first to point out that lots of solutions of Diophantine equations implicitly used unique factorisation in whatever ring the equation in question asked for and that it didn't necessarily hold? And didn't Dedekind, inspired by Kummer's "ideal numbers", introduce the notion of an ideal and of ideal class group? I'm not nearly as knowledgeable in the history if mathematics as you are, so I am quite intrigued by your comment.

Is it possible that you meant $a_i^2−b_i^2D=x_i^3$ or $a_i^2+b_i^2|D|=x_i^3$, and accordingly for the following formulae?

That rule of thumb sounds like it could be very close to the truth and I think I will adopt it. I guess I will leave this one, since your average undergraduate doesn't usually know Dirichlet's unit theorem (not the undergrads I know, anyway), so probably wouldn't get away with using it in a solution.

Just saw Kevin's comment. Maybe I should withdraw my post. What do you think? On the other hand, if it's homework, I doubt that this is the sort of answer the lecturer is looking for.

I am pretty sure that this won't be quite so easy for arbitrary $p$. As Remke mentions in his answer, arbitrarily large $p$-Selmer groups over $\mathbb{Q}$ are only known to exist for a few specific values of $p$ and more or less each one of them has been worth a publication.