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On the number of solutions of simultaneous Pell equations. II. (English) Zbl 1165.11034

The study of simultaneous Pell equations is well known and has a very rich history. Determining the number of solutions is a problem that recently progresses. In 1998, M. A. Bennett [J. Reine Angew. Math. 498, 173–199 (1998; Zbl 1044.11011)] showed that the general simultaneous Pell equations \(y^2-az^2=1\) and \(x^2-bz^2=1\) possess at most three solutions in positive integers \(x,\, y,\, z\). In 2002, P. Z. Yuan [Acta Arith. 101, No. 3, 215–221 (2002; Zbl 1003.11007)] improved Bennett’s result by proving that these simultaneous Pell equations have at most two solutions in positive integers \(x, y, z\), for \(\max\{a, b\} > 1.4\cdot 10^{57}\).
In this paper, the authors remove the above condition imposed by Yuan on \(a\) and \(b\). In fact, they prove that the system \(y^2-az^2=1,\) \(x^2-bz^2=1\) has at most two solutions in positive integers \(x, y, z\). For the proof, the authors use some results obtained by Yuan [loc. cit.], an improvement of a gap principle for the solutions due to the first author [loc. cit.], and a result of E. M. Matveev [Izv. Math. 64, No. 6, 1217–1269 (2000); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 64, No. 6, 125–180 (2000; Zbl 1013.11043)] on the lower bounds of linear forms in logarithms.

MSC:

11D45 Counting solutions of Diophantine equations
11D09 Quadratic and bilinear Diophantine equations
11D25 Cubic and quartic Diophantine equations
11J13 Simultaneous homogeneous approximation, linear forms
11J86 Linear forms in logarithms; Baker’s method
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