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POWERS FROM PRODUCTS OF CONSECUTIVE TERMS IN ARITHMETIC PROGRESSION

Published online by Cambridge University Press:  20 February 2006

M. A. BENNETT ,
N. BRUIN ,
K. GYÖRY  and
L. HAJDU
M. A. BENNETT
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2 [email protected]
N. BRUIN
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6 [email protected]
K. GYÖRY
Affiliation:
Number Theory Research Group of the Hungarian Academy of Sciences, Institute of Mathematics, University of Debrecen, P.O. Box 12, 4010 Debrecen, Hungary, [email protected], [email protected]
L. HAJDU
Affiliation:
Number Theory Research Group of the Hungarian Academy of Sciences, Institute of Mathematics, University of Debrecen, P.O. Box 12, 4010 Debrecen, Hungary, [email protected], [email protected]
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Abstract

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.

Keywords

11D6111B25
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 92 , Issue 2 , March 2006 , pp. 273 - 306
Copyright
2006 London Mathematical Society

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Footnotes

Research supported in part by grants from NSERC (M.A.B. and N.B.), the Erwin Schrödinger Institute in Vienna (M.A.B. and K.G.), the Netherlands Organization for Scientific Research (NWO) (K.G. and L.H.), the Hungarian Academy of Sciences (K.G. and L.H.), by FKFP grant 3272-13/066/2001 (L.H.) and by grants T29330, T42985 (K.G. and L.H.), T38225 (K.G.) and F34981 (L.H.) of the Hungarian National Foundation for Scientific Research.