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Catalan's Conjecture: Another old Diophantine problem solved

Author(s): Tauno Metsänkylä
Journal: Bull. Amer. Math. Soc. 41 (2004), 43-57.
MSC (2000): Primary 11D41, 00-02; Secondary 11R18
Posted: September 5, 2003
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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.

References:

[ ]

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Additional Information:

Tauno Metsänkylä
Affiliation: Department of Mathematics, University of Turku, FIN-20014 Turku, Finland
Email: taumets@utu.fi

DOI: 10.1090/S0273-0979-03-00993-5
PII: S 0273-0979(03)00993-5
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
Posted: September 5, 2003
Copyright of article: Copyright 2003, American Mathematical Society

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