On arithmetic progressions on Pellian equations

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,...

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Bibliographic Details
Published in:Acta mathematica Hungarica 2008-07, Vol.120 (1-2), p.29-38
Main Authors: Dujella, A., Pethő, A., Tadić, P.
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
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Summary: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.
ISSN:0236-5294
1588-2632
DOI:10.1007/s10474-007-7087-1