The ABC-Conjecture: Beyond The Fermat Last Theorem

Hossein Teimoori Faal

Abstract

Fermat's last theorem, involves an equation of the form x^n + y^n = z^n. More than 300 years ago, Pierre de Fermat (1601-1665) conjectured that the equation has no solution if x, y, and z are all positive integers and n is a whole number greater than 2. Finally, in May 1995, two mathematicians proved the Fermat Last Theorem.

Interestingly, the proof of Fermat's last theorem was a by-product of a deep inroads into proving the Shimura-Taniyama-Weil conjecture. Now, the Wiles effort could help point the way to a general theory of three-variable Diophantine equations.

The key element appears to be a problem termed the ABC conjecture, which was formulated in the mid-1980s by Joseph Oesterle of the University of Paris VI and David W. Masser of the Mathematics Institute of the University of Basel in Switzerland. That conjecture offers a new way of expressing Diophantine problems, in effect