Dujella, A.; Pethő, A.; Tadić, P. On arithmetic progressions on Pellian equations. (English) Zbl 1174.11033 Acta Math. Hung. 120, No. 1-2, 29-38 (2008). 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. Reviewer: Lajos Hajdu (Debrecen) Cited in 1 ReviewCited in 5 Documents MSC: 11D09 Quadratic and bilinear Diophantine equations 11B25 Arithmetic progressions 11G05 Elliptic curves over global fields 11Y50 Computer solution of Diophantine equations Keywords:Pell equation; arithmetic progression; elliptic curve Citations:Zbl 1142.11016 Software:ecdata; APECS PDFBibTeX XMLCite \textit{A. Dujella} et al., Acta Math. Hung. 120, No. 1--2, 29--38 (2008; Zbl 1174.11033) Full Text: DOI Link References: [1] C. Batut, D. Bernardi, H. Cohen and M. Olivier, GP/PARI, Université Bordeaux I (1994). [2] I. Connell, APECS, ftp://ftp.math.mcgill.ca/pub/apecs/ [3] J. E. Cremona, Algorithms for Modular Elliptic Curves, Cambridge University Press (Cambridge, 1997). · Zbl 0872.14041 [4] A. Dujella, An extension of an old problem of Diophantus and Euler. II, Fibonacci Quart., 40 (2002), 118–123. · Zbl 1125.11308 [5] A. Dujella, An example of elliptic curve over \(\mathbb{Q}\) with rank equal to 15, Proc. Japan Acad. Ser. A Math. Sci., 78 (2002), 109–111. · Zbl 1040.11040 · doi:10.3792/pjaa.78.109 [6] R. Miranda, An overview of algebraic surfaces, in: Algebraic Geometry (Ankara, 1995), Lecture Notes in Pure and Appl. Math. 193, Dekker (New York, 1997), pp. 157–217. · Zbl 0903.14011 [7] I. Niven, Diophantine Approximations, Wiley (New York, 1963). · Zbl 0115.04402 [8] A. Petho and V. Ziegler, Arithmetic progressions on Pell equations, to appear in J. Number Theory. [9] T. Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Pauli, 39 (1990), 211–240. · Zbl 0725.14017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.