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Catalan's Conjecture: Another old Diophantine problem solved

Author: Tauno Metsänkylä
Journal: Bull. Amer. Math. Soc. 41 (2004), 43-57
MSC (2000): Primary 11D41, 00-02; Secondary 11R18
Published electronically: September 5, 2003
MathSciNet review: 2015449
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Abstract: Catalan's Conjecture predicts that 8 and 9 are the only consecutive perfect powers among positive integers. The conjecture, which dates back to 1844, was recently proven by the Swiss mathematician Preda Mihailescu. A deep theorem about cyclotomic fields plays a crucial role in his proof.

Like Fermat's problem, this problem has a rich history with some surprising turns. The present article surveys the main lines of this history and outlines Mihailescu's brilliant proof.

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Additional Information

Tauno Metsänkylä
Affiliation: Department of Mathematics, University of Turku, FIN-20014 Turku, Finland

Keywords: Catalan's Conjecture, Diophantine equations of higher degree, cyclotomic fields, research exposition
Received by editor(s): March 5, 2003
Received by editor(s) in revised form: July 14, 2003
Published electronically: September 5, 2003
Article copyright: © Copyright 2003 American Mathematical Society