Solutions of a class of quartic Thue inequalities

Abstract

Let c be an integer such that \(c\ge 3\). In this paper, we use the method of Tzanakis to transform the quartic Thue equation $$\begin{aligned} x^4 +4(c^2-1) x^3y +(8c^2+6) x^2 y^2 +4(c^2-1) xy^3 +y^4 = \mu \end{aligned}$$into a systems of Pell equations. Then, we determine all positive integer solutions (x, y) with \(0<|\mu |\le 2c\).