6 The ABC ’ s of Number Theory

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 Oesterlé. 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 + y = z. 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.‡ 6.1 Pythagorean triples: x2 + y2 = z2 An ordered triple (x, y, z) of integers is called a Pythagorean triple if and only if it solves the Diophantine equation x + y = z; that is, if and only if |x| and |y| are the lengths of the sides, and |z| the length of the hypotenuse, of a right triangle. (We allow degenerate triangles with a “side” of length zero.) It is well-known that every such triple is proportional to (x, y, z) = (m − n, 2mn,m + n) (6.1) for some integers m,n. Equivalently (dividing by n to obtain polynomials in the single rational variable t = m/n), the solution (x, y, z) is proportional to (t − 1, 2t, t + 1) for some t ∈ Q, or to (1, 0, 1) which arises for “t =∞” (corresponding to (m,n) = (1, 0)). That is, all Pythagorean triples are accounted for by the single polynomial identity (t − 1) + (2t) = (t + 1). (6.2) †Noam D. Elkies earned his doctorate in mathematics in 1987 at Harvard, where his advisors where Professors Barry Mazur and Benedict H. Gross. After three years in Harvard’s Society of Fellows he joined the Mathematics faculty and has remained at Harvard since. Most of his research is in number theory, usually Diophantine geometry (the combination of algebraic geometry and Diophantine equations) and/or computational number theory. Other interests include some combinatorial mathematics (lattices and codes, incidence geometry, and combinatorial games) and, outside of mathematics, classical music (mostly composition and piano) and chess (usually chess problems and endgames). ‡Supported in part by NSF grant DMS-0501029.