On simultaneous Pell equations and related Thue equations

Abstract

In this paper, we prove that the simultaneous Pell equations \[ x 2 − ( m 2 − 1 ) y 2 = 1 , z 2 − ( n 2 − 1 ) y 2 = 1 x^2-(m^2-1)y^2=1,\, z^2-(n^2-1)y^2=1 \] have only a positive integer solution ( x , y , z ) = ( m , 1 , n ) (x, y, z) = (m, 1, n) if m > n ≤ m + m ε , 0 > ε > 1 m > n \le m+m^{\varepsilon }, 0 > \varepsilon > 1 and m ≥ 202304 1 1 − ε m \ge 202304^{\frac {1}{1-\varepsilon }} . Using a computational reduction method we can omit the lower bound for m m when m > n ≤ m 1 5 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.