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On arithmetic progressions on Pellian equations. (English) Zbl 1174.11033

Let \(d,m\) be integers, \(d\) is not a square. The authors consider arithmetic progressions in the \(y\)-component of the solutions of the Pellian equation \(x^2-dy^2=m\). They prove that up to the exceptions \((0,1,2,3)\) and \((-3,-2,-1,0)\) for any four-term arithmetic progression there exist infinitely many \(d,m\) with the prescribed properties such that the members of the progression occur as a \(y\)-component of the above Pellian equation. Further, they give several five-term progressions arising from the \(y\)-components of solutions of two such equations simultaneously. Finally, they give a particular example for \(d,m\) when the corresponding Pellian equation has a seven-term arithmetic progression in the \(y\)-components of its solutions. To prove their results, the authors study a parametric family of elliptic equations. The results fit well to related theorems of A. Pethő and V. Ziegler[J. Number Theory, 128, No. 6, 1389–1409 (2008; Zbl 1142.11016)], obtained previously.

MSC:

11D09 Quadratic and bilinear Diophantine equations
11B25 Arithmetic progressions
11G05 Elliptic curves over global fields
11Y50 Computer solution of Diophantine equations

Citations:

Zbl 1142.11016

Software:

ecdata; APECS
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Full Text: DOI Link

References:

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