Une question sur les équations xm ? ym = Rzn (A Question on the Equations xm ? ym = Rzn) |
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Authors: | Alain Kraus |
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Institution: | (1) Institut de Mathématiques, UMR 7586 du CNRS, Équipe de Théorie des Nombres, Université de Paris VI, 175 Rue du Chevaleret, PARIS, 75013, France |
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Abstract: | Let m and n be integers at least two and R be a nonzero natural number. In this paper, we study the problem of the determination of the proper solutions of the Diophantine equation x
m
– y
m
equals; Rz
n
. We raise a question concerning the existence of any proper nontrivial solution of this equation, in case some precise conditions are satisfied by the triple (m, n, R). We prove some results about it. |
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Keywords: | generalised Fermat equations cyclotomic polynomials rational points elliptic curves Dem'janenko's method |
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