Catalan’s conjecture states that the equation xp−yq=1 admits the unique solution 32−23=1 in integers x,y,p,q≥2. The conjecture has been proved by Mihăilescu in 2002 using the theory of cyclotomic fields and Galois modules. Here, relying only on elementary methods, we prove several instances of this result. In particular, we show it in the following cases: p even, q is even, x divides q, y divides x−1, y is a power of a prime, and y≤pp/2.

On consecutive perfect powers with elementary methods

Leonetti P.
2021-01-01

Abstract

Catalan’s conjecture states that the equation xp−yq=1 admits the unique solution 32−23=1 in integers x,y,p,q≥2. The conjecture has been proved by Mihăilescu in 2002 using the theory of cyclotomic fields and Galois modules. Here, relying only on elementary methods, we prove several instances of this result. In particular, we show it in the following cases: p even, q is even, x divides q, y divides x−1, y is a power of a prime, and y≤pp/2.
2021
978-3-030-67995-8
978-3-030-67996-5
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2146531
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