Glasnik Matematicki, Vol. 48, No. 1 (2013), 1-22.

ON GEOMETRIC PROGRESSIONS ON PELL EQUATIONS AND LUCAS SEQUENCE

Attila Bérczes and Volker Ziegler

Institute of Mathematics, University of Debrecen, Number Theory Research Group, Hungarian Academy of Sciences and University of Debrecen, H-4010 Debrecen, P.O. Box 12, Hungary
e-mail: berczesa@math.klte.hu

Institute for Analysis and Computational Number Theory, Graz University of Technology, Steyrergasse 30/IV, A-8010 Graz, Austria
e-mail: ziegler@finanz.math.tugraz.at


Abstract.   We consider geometric progressions on the solution set of Pell equations and give upper bounds for such geometric progressions. Moreover, we show how to find for a given four term geometric progression a Pell equation such that this geometric progression is contained in the solution set. In the case of a given five term geometric progression we show that at most finitely many essentially distinct Pell equations exist, that admit the given five term geometric progression. In the last part of the paper we also establish similar results for Lucas sequences.

2010 Mathematics Subject Classification.   11D09, 11B25, 11G05.

Key words and phrases.   Pell equations, geometric progressions, elliptic curves.


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DOI: 10.3336/gm.48.1.01


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