The ABC's of Number Theory

Abstract

The ABC conjecture is a central open problem in modern number theory, connecting results, techniques and questions ranging from elementary number theory and algebra to the arithmetic of elliptic curves to algebraic geometry and even to entire functions of a complex variable. The conjecture asserts that, in a precise sense that we specify later, if (A,B,C) are relatively prime integers such that (A + B = C) then (A,B,C) cannot all have many repeated prime factors. This expository article outlines some of the connections between this assertion and more familiar Diophantine questions, following (with the occasional scenic detour) the historical route from Pythagorean triples via Fermat’s Last Theorem to the formulation of the ABC conjecture by Masser and Oesterl´e. We then state the conjecture and give a sample of its many consequences and the few very partial results available. Next we recite Mason’s proof of an analogous assertion for polynomials (A(t),B(t), C(t)) that implies, among other things, that one cannot hope to disprove the ABC conjecture using a polynomial identity such as the one that solves the Diophantine equation (x^2 + y^2 = z^2). We conclude by solving a Putnam problem that predates Mason’s theorem but is solved using the same method, and outlining some further open questions and fragmentary results beyond the ABC conjecture.