Powers from products of consecutive terms in arithmetic progression

• Autores: Michael A. Bennett, Nils Bruin, Kálmán Györy, Lajos Hajdu
• Localización: Proceedings of the London Mathematical Society, ISSN 0024-6115, Vol. 92, Nº 2, 2006, págs. 273-306
• Idioma: inglés
• DOI: 10.1112/s0024611505015625
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• Resumen

  • We show that if $k$ is a positive integer, then there are, under certain technical hypotheses, only finitely many coprime positive $k$-term arithmetic progressions whose product is a perfect power. If $4 \leq k \leq 11$, we obtain the more precise conclusion that there are, in fact, no such progressions. Our proofs exploit the modularity of Galois representations corresponding to certain Frey curves, together with a variety of results, classical and modern, on solvability of ternary Diophantine equations. As a straightforward corollary of our work, we sharpen and generalize a theorem of Sander on rational points on superelliptic curves.