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On the Diophantine Equation $n(n\,+\,d)\,\cdots \,(n\,+\,(k\,-\,1)d)\,=\,b{{y}^{l}}$

Published online by Cambridge University Press:  20 November 2018

K. Győry ,
L. Hajdu  and
N. Saradha
K. Győry
Affiliation:
Number Theory Research Group of the Hungarian Academy of Sciences, and Institute of Mathematics University of Debrecen P.O. Box 12 4010 Debrecen Hungary, e-mail: gyory@math.klte.hu
L. Hajdu
Affiliation:
Number Theory Research Group of the Hungarian Academy of Sciences, and Institute of Mathematics University of Debrecen P.O. Box 12 4010 Debrecen Hungary, e-mail: hajdul@math.klte.hu
N. Saradha
Affiliation:
School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Mumbai 400 005 India, e-mail: saradha@math.tifr.res.in
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Abstract

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We show that the product of four or five consecutive positive terms in arithmetic progression can never be a perfect power whenever the initial term is coprime to the common difference of the arithmetic progression. This is a generalization of the results of Euler and Obláth for the case of squares, and an extension of a theorem of Győry on three terms in arithmetic progressions. Several other results concerning the integral solutions of the equation of the title are also obtained. We extend results of Sander on the rational solutions of the equation in $n,\,y$ when $b\,=\,d\,=\,1$. We show that there are only finitely many solutions in $n,\,d,\,b,\,y$ when $k\,\ge \,3,\,l\,\ge \,2$ are fixed and $k\,+\,l\,>\,6$.

Keywords

11D41
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 47 , Issue 3 , 01 September 2004 , pp. 373 - 388
Copyright
Copyright © Canadian Mathematical Society 2004

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