On simultaneous Pell equations and related Thue equations

Abstract:

In this paper, we prove that the simultaneous Pell equations \[ x^2-(m^2-1)y^2=1, z^2-(n^2-1)y^2=1 \] have only a positive integer solution $(x, y, z) = (m, 1, n)$ if $m < n \le m+m^{\varepsilon }, 0 < \varepsilon < 1$ and $m \ge 202304^{\frac {1}{1-\varepsilon }}$. Using a computational reduction method we can omit the lower bound for $m$ when $m<n\le m^{\frac {1}{5}}$. Moreover, we apply our main result to a family of Thue equations in two parameters studied by Jadrijević.