profinite topology

Recently I've been using something similar to Mordell-Weil sieve to study certain Diophantine equations. This has been pretty successful, which is wonderful. And I've been wondering, if these are successful since I am lucky, I am not unlucky, or because this is just going to work. Here is a bit more clear explanation (I'm going to focus in 1-d version).

We can actually abstract the sieving problem I'm facing with in the following way. Say for any prime $p$, we have a set $S_p \subset \ZZ$, and we have that $\bigcup S_p = \ZZ$. Can we find a finite subset of primes so that $\bigcup S_p = \ZZ$? In the particular case that I'm faced with the sets $S_p = \cup (a+m_p \ZZ)$ with $m_p>0$ (in fact, $m_p=\#E(\FF_p)$). Will there always be a finite subcover in here? This, of course, will be true if $\ZZ$ is compact under a certain topology for integers (I think it is called profinite topology). I'm guessing that $\ZZ$ is not compact under profinite topology though (otherwise, that would make life a lot nicer). However, I am still unsure on weather or not in my particular case, these particular open coverings of $\ZZ$ have a finite subcover or not.

y^2=x^3+Dz^12,

I have my student working on finding primitive integer solutions to some Diophantine equations of the form $C:y^2=x^3+Dz^{12}$. This is a genus two curve (in the appropriate weighted projective space), and it is a double cover of an elliptic curve, so computation of the Mordell-Weil rank is usually pretty straightforward. Say we find a $D$ so that the elliptic curve $y^2=x^3+D$ has rank $2$, but we suspect that there are no non-primitive solutions there. Can we prove this? I suspect that a simple a Mordell-Weil sieving should give us the desired result, and it seems pretty successful so far. However, it seems the trivial solutions
$(1,1,0)$ is causing some trouble for us. Right now, we're trying to figure out if there is a way of getting pass this problem.