The ABC Conjecture
by
Kevin Broughan
Dept of Mathematics, University of Waikato

The ABC conjecture arose in a discussion between Masser and Oesterlè in 1985. It is a simple but powerful relationship between the additive and the multiplicative properties of integers. If two numbers with no common factors are each divisible by relatively large high powers of integers then their sum tends not to be itself divisible by a large high power. This phenomena is revealed in Fermat's Last Theorem and the finiteness of the number of solutions to the Catalan Problem. The square-free part of an integer n is the largest square free number which divides it. In other words it is the product of each of the primes which divide n. This so-called square-free core, conductor or radical is therefore:

N(n)= Õ p|n  p

Let a+b=c and (a, b)=1 and all be positive integers. For most choices of these numbers, N(abc) is greater than c, i.e. \fracN(abc)c > 1. However Masser proved that the ratio can be made arbitrarily small. The ABC conjecture states that if we raise the numerator to a power strictly greater than 1, then the ratio is bounded away from zero, by a constant dependent on the power (the larger the power the bigger the constant). In other words for every \epsilon > 0 there is a k_\epsilon > 0 such that

c < k\epsilon N(abc)1+\epsilon

whenever the positive integers a, b, c are such that (a, b)=1 and a+b=c.

In this lecture I will give some examples of small values for the ratio \fracN(abc)c, (Note that 2+3¹⁰ 109=23⁵ gives a nice value), demonstrate how ABC implies the asymptotic Fermat Theorem, namely there exists an integer n_o such that

xn+yn=zn

has no solutions in integers x, y, z for n > n_o, the asymptotic Catalan conjecture (also known to be true) that

xn-ym=1

has at most a finite number of solutions, use it to explore the Diophantine equation

n!+1=m2

and sumarise (if time permits) a number of advanced applications from a lecture on the Conjecture given by Dorian Goldfeld at the Institute for Advanced Study, Princeton in 1999.

References:

[1] I. Petersen, http://www.maa.org/mathland/mathtrek_12_8.html

[2] S. Lang, Old and new conjectured diophantine inequalities. Bull. Amer. Math. Soc. 23 (1990), 37-75.

[3] A. Nitaj, La conjecture abc. Enseignement Math. 42 (1996), 3-24.

Date received: October 18, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caek-70.