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Perfect powers from products of consecutive terms in arithmetic progression

Part of: Diophantine equations Sequences and sets

Published online by Cambridge University Press:  01 July 2009

K. Győry ,
L. Hajdu  and
Á. Pintér
K. Győry
Affiliation:
Institute of Mathematics, University of Debrecen, and the Number Theory Research Group of the Hungarian Academy of Sciences, PO Box 12, H-4010 Debrecen, Hungary (email: gyory@math.klte.hu)
L. Hajdu
Affiliation:
Institute of Mathematics, University of Debrecen, and the Number Theory Research Group of the Hungarian Academy of Sciences, PO Box 12, H-4010 Debrecen, Hungary (email: hajdul@math.klte.hu)
Á. Pintér
Affiliation:
Institute of Mathematics, University of Debrecen, and the Number Theory Research Group of the Hungarian Academy of Sciences, PO Box 12, H-4010 Debrecen, Hungary (email: apinter@math.klte.hu)
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Abstract

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We prove that for any positive integers x,d and k with gcd (x,d)=1 and 3<k<35, the product x(x+d)⋯(x+(k−1)d) cannot be a perfect power. This yields a considerable extension of previous results of Győry et al. and Bennett et al., which covered the cases where k≤11. We also establish more general theorems for the case where x can also be a negative integer and where the product yields an almost perfect power. As in the proofs of the earlier theorems, for fixed k we reduce the problem to systems of ternary equations. However, our results do not follow as a mere computational sharpening of the approach utilized previously; instead, they require the introduction of fundamentally new ideas. For k>11, a large number of new ternary equations arise, which we solve by combining the Frey curve and Galois representation approach with local and cyclotomic considerations. Furthermore, the number of systems of equations grows so rapidly with k that, in contrast with the previous proofs, it is practically impossible to handle the various cases in the usual manner. The main novelty of this paper lies in the development of an algorithm for our proofs, which enables us to use a computer. We apply an efficient, iterated combination of our procedure for solving the new ternary equations that arise with several sieves based on the ternary equations already solved. In this way, we are able to exclude the solvability of the enormous number of systems of equations under consideration. Our general algorithm seems to work for larger values of k as well, although there is, of course, a computational time constraint.

Keywords

perfect powersarithmetic progressionternary diophantine equationsmodular forms

MSC classification

Secondary: 11D61: Exponential equations 11B25: Arithmetic progressions
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 4 , July 2009 , pp. 845 - 864
Copyright
Copyright © Foundation Compositio Mathematica 2009

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