Solving a family of quartic Thue inequalities using continued fractions
In the paper under review the authors consider the Thue inequality \[ |x^4+2(1-n^2)x^2y^2+y^4|\leq 2n+3 \] and prove that for \(n\geq 3\) the only primitive solutions are \((0,\pm 1)\) and \((\pm 1,0)\), if \(2(n^2-1)\) is not a perfect square. It has the additional primitive solutions \((\pm 1,\pm \nu)\) and \((\pm \nu,\pm 1)\) only if \(2(n^2-1)=\nu^2\). The authors obtain their result by applying \textit{N. Tzanakis} method [Acta Arith. 64, No.3, 271--283 (1993; Zbl 0774.11014)] and using a generalization of \textit{R. T. Worley}'s theorem on Diophantine approximations [J. Aust. Math. Soc., Ser. A 31, 202--206 (1981; Zbl 0465.10026)].
scientific article; zbMATH DE number 5940453