NUMBER OF SOLUTIONS FOR QUARTIC SIMPLE THUE EQUATIONS

Abstract

Let F(X, Y) = bX4 - aX3Y - 6bX2Y2 + aXY3 + bY4 ∈ Z[X, Y]. We show that the number of solutions for the Thue equation F(x, y) = ±1 is 0 or 4 except for a few already known cases. To obtain an upper bound for the size of solutions, we use Padé approximation method. To obtain a lower bound for the size of solutions, we construct a continued fraction with positive or negative rational partial quotients. This construction is carried out carefully by using special properties of the form F. Combining these lower and upper bounds, we obtain the result.