On a class of quartic Thue equations with three parameters

Abstract

Let k, m and n be integers. In this paper, for a fixed integer μ nonzero, we show that the family of Thue equation x^4 − kmnx^3y + (km^2 − kn^2 + 2)x^2y^2 + kmnxy^3 + y^4 = μ, is reducible by Tzanakis’s method into a system of pellian equations kV^2 − (km^2 + 4)U^2 = −4μ; kZ^2 − (kn^2 − 4)U^2 = 4μ, with any triple of integers (k, m, n) such that k > 0, |n| ≥ 2, |m| ≥ 2. We consider this system for any even integer k = 2, μ = 1 and we prove that for all integers |n| ≥ 2 and |m| ≥ 2 that are sufficiently large and have sufficiently large common divisor this system has only the trivial solutions (U, V, Z, ) =(±1, ±m, ±n). We also show that if k = 2 is even, then the system has in general at most 8 solutions in positive integers.