Methods for “additive” problems in number theory

Question

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?

Answer 1

There's the recent XYZ conjecture 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 are quite informative on these matters.

Answer 2

I'm currently studying with Melvyn Nathanson,who is really considered one of the experts on additive number theory.His texts,ADDITIVE NUMBER THEORY:THE CLASSICAL BASES and ADDITIVE NUMBER THEORY:INVERSE PROBLEMS are really the standard introductions to the subject.They are both published by Springer-Verlag.

I'd also look at his papers on the Archive-he's written many of them on open problems in additive number theory.A full list can be found-with links to many of them in PDF for download-at: http://front.math.ucdavis.edu/search?a=Nathanson%2C+Melvyn&t=&q=&c=&n=40&s=Listings.

You'll also find the most recent version of an overview of open problems in both additive number theory and combinatorics that Melvyn's been working on for a few years at:

http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.2073v1.pdf

I think you'll find the latter reference particularly pertinent to your questions.

The subject is very interesting since it essentially involves all subsets of the integers whose members can be expressed as arithmetic progressions.This provides connections to not only number theoretic questions in geometric group theory and analysis,but I'm currently investigating the role of topology in determining the structure of such "sumsets" of Z.