A family of quartic Thue inequalities

We prove that the only primitive solutions of the Thue inequality |x^4-4cx^y+(6c+2)x^2y^2+4cxy^3+y^4| =4 is an integer, are (x, y)=(+-1, 0), (0, +-1), (1, +-1), (-1, +-1), (+-1, -+2), (+-2, +-1). Solving Thue equations F(x, y)=m of the special type, using the method of Tzanakis, reduces to solving the system of Pellian equations. The application of Tzanakis method for solving Thue equations has several advantages. We show that some additional advantages appear when one deals with corresponding Thue inequalities. Namely, the theory of continued fractions can be used in order to determine small values of m for which the equation F(x, y)=m has a solution. In particular, we use characterization in terms of continued fractions of alpha of all fractions a/b satisfying the inequality |alpha - a/b| < 2/b^2.

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