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The Amazing ABC Conjecture

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Author:Ivars Peterson (MathTrek)
Description: In number theory, straightforward, reasonable questions are remarkably easy to ask, yet many of these questions are surprisingly difficult or even impossible to answer. Fermat's last theorem, for instance, 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. Andrew J. Wiles of Princeton University finally proved Fermat's conjecture in 1994... the Wiles proof of Fermat's last theorem was a by-product of his 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. Historically, mathematicians have always had to state and solve such problems on a case-by-case basis. An overarching theory would represent a tremendous advance. 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 translating an infinite number of Diophantine equations (including the equation of Fermat's last theorem) into a single mathematical statement.

Levels: Middle School (6-8), High School (9-12), College
Languages: English
Resource Types: Articles
Math Topics: Number Theory

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