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On arithmetic progressions on Pellian equations

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Abstract

We consider arithmetic progressions consisting of integers which are y-components of solutions of an equation of the form x 2 − dy 2 = m. We show that for almost all four-term arithmetic progressions such an equation exists. We construct a seven-term arithmetic progression with the given property, and also several five-term arithmetic progressions which satisfy two different equations of the given form. These results are obtained by studying the properties of a parametric family of elliptic curves.

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Authors and Affiliations

  1. Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000, Zagreb, Croatia

    A. Dujella & P. Tadić

  2. Faculty of Informatics, University of Debrecen, H-4010, Debrecen, P.O. Box 12, Hungary

    A. Pethő

Authors
  1. A. Dujella
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  2. A. Pethő
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  3. P. Tadić
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Correspondence to A. Dujella.

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Dujella, A., Pethő, A. & Tadić, P. On arithmetic progressions on Pellian equations. Acta Math Hung 120, 29–38 (2008). https://doi.org/10.1007/s10474-007-7087-1

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  • DOI: https://doi.org/10.1007/s10474-007-7087-1

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