Powers from Products of Consecutive Terms in Arithmetic Progression
Powers from Products of Consecutive Terms in Arithmetic Progression
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DOI：
10.1112/s0024611505015625
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发表时间：
2006-03
影响因子：
1.8
通讯作者：
M. Bennett;Nils Bruin;K. Gyory;L. Hajdu
中科院分区：
文献类型：
--
作者：
M. Bennett;Nils Bruin;K. Gyory;L. Hajdu
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 ⩽ k ⩽ 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. 2000 Mathematics Subject Classification 11D61, 11B25
