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A new criterion for the Catalan equation. (Un nouveau critère pour l’équation de Catalan.) (French) Zbl 1008.11011

This paper contains the following new criterion for the Catalan equation \[ x^p-y^q=\pm 1 \] where \(p\) and \(q\) are odd primes and \(x\), \(y\) are positive rational integers.
Theorem. Let \(p\) and \(q\) be two odd prime numbers. If there exist positive rational integers \(x\) and \(y\) such that \(|x^p-y^q|=1\), then \(q\) divides \(h_p^-\), the relative class-number of the cyclotomic field \({\mathbb Q}(e^{2i\pi/p})\).
The proof is elementary and makes an important use of cyclotomy. It has some links with previous works of Bilu and Bilu-Hanrot. As a corollary, the authors prove that \(\min{p,q}\geq 43\); this is the first proof of such a result without any use of lower bounds of linear forms in logarithms.
In his papers of 1999 and 2002, which finish the proof of Catalan’s conjecture, P. Mihăilescu explains that this short paper had a great influence on him and showed him the importance of cyclotomy methods to solve Catalan’s equation.

MSC:

11D61 Exponential Diophantine equations
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