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.


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
Full Text: DOI