## 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

### Citations:

Zbl 1044.11011; Zbl 1003.11007; Zbl 1013.11043
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