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Diophantine Equations

In: Mathematics Unlimited — 2001 and Beyond

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  • Gerd Faltings

Abstract

This will be a short introduction to what I know about diophantine equations. Since Fermat this topic has fascinated the best minds in mathematics, as well as many amateurs. Progress resulted not just from bettering one’s predecessors in what they did, but also from the introduction of new methods and insights. Kummer’s introduction of ideal numbers, Weil’s invention of abstract algebraic varieti es, and the recent introduction of modular elliptic curves are examples of this. This makes it difficult to predict the future, but since this is the main topic of this volume, I will still try to make some remarks about it. Anyway the difficulty of predicting has not deterred experts from trying and sometimes failing. Hilbert’s Paris problems and his estimate about the difficulty of the Riemann hypothesis, respectively proving the transcendence of 2√2, are famous examples of success and failure. We leave it to future readers to locate our effort between these two benchmarks.

Suggested Citation

  • Gerd Faltings, 2001. "Diophantine Equations," Springer Books, in: Björn Engquist & Wilfried Schmid (ed.), Mathematics Unlimited — 2001 and Beyond, pages 449-454, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-56478-9_21
    DOI: 10.1007/978-3-642-56478-9_21
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