I've read an interesting article, [math.NT/0409456](http://arxiv.org/abs/math.NT/0409456) where you're just trying to solve a simple problem:



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> For a given (finite) set of primes S find all solutions to an equation `a + b = c` with the condition that all prime divisiors of integers a, b, c must be in S.
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and this problem turns out to be very geometric. It turns out (and I tell that in comments below) you're actually dealing with sections of certain projective morphism of schemes `R --> Spec ZZ \ S`. The article then proves that the number of solutions to the equation is finite by proving that the number of these sections is finite.


Is there kind of general theory or other methods to prove things about sections of these maps? What is the intuition used here? Would there be a way to count these solutions?



Solving "a, b, a+b have given divisors" problem
This is bound to be pretty hard! Knowing the prime divisors of a,b,c implies knowing the radical rad(abc) (the product of the distinct primes dividing abc), and then knowing the solutions a,b,c of a + b = c in positive integers, you would know the largest c with a,b,c coprime (pairwise or not, doesn't matter). So you would be in a position to settle the ABC conjecture c \leq C(epsilon)rad(abc)^{1 + \epsilon} if you could handle all those S-unit equations. The ABC conjecture is *the* outstanding problem in Diophantine analysis.


