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≠0, we show that the family of Thue equationx4−kmnx3y+(km2−kn2+2)x2y2+kmnxy3+y4=μ,is reducible by Tzanakis's method into a system of pellian equationskV2−(km2+4)U2=−4μ,kZ2−(kn2−4)U2=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.