I was wondering if there are some classical methods to tackle problems in number theory dealing with sums where the primes are not well-"controled". I talk about problems where we want to link a certain sum with information about the primes dividing the elements of the sum: the $abc$ conjecture is an example of such a problem, since we want to link $a$, $b$ and $a+b$ to the prime factors of $abc$, knowing that $a$, $b$ and $a+b$ are coprime. Another "additive" problem is the Goldbach conjecture.


Since the natural way to deal with prime factors is for... factorization, these kinds of problems look way more complicated. Except sieve methods, are there any conclusive methods?



Methods for "additive" problems in number theory
There's the recent [XYZ conjecture](http://arxiv.org/pdf/0911.4147) of Lagarias and Soundararajan. It concerns bounding $\log(\log(A+B))$ in terms of the largest prime $p$ dividing $AB(A+B).$


Also, a great way to understand properties a triple $(A,B,C)$ of nonzero integers satisfying
$$A + B = C$$
is to consider properties of the Frey elliptic curve
$$ E\_{(A,B,C)} : y^2 = x (x-A) (x+B)$$
as well as additional structures (e.g. modular forms, Galois representations) associated with this elliptic curve. This is one of the routes by which the ABC conjecture was originally discovered.


Barry Mazur's notes [here](http://www.math.harvard.edu/~mazur/papers/scanQuest.pdf) are quite informative on these matters.


