A parametric family of cubic Thue equations
O. Lecacheux in 1993 and L. C. Washington in 1997 have dealt with the parametric Thue equation (*) \(\Phi_n(x,y)= x^3-(n^3-2n^2+3n-3)\cdot x^2y-n^2xy^2-y^3= \pm1\). The purpose of this paper is the complete solution of equation (*) for all integer parameters \(n\geq 1\). It is proved that, for all \(n\geq 1\), \(n\neq 2\), the only integer solutions to equation (*) are given by \(\pm(x,y)=(1,0),(0,1)\). Let \(K_n\) be the field generated by a root of the polynomial \(\phi_n(x)=\Phi_n(x,1)\). The extension \(K_n/Q\) is Galois and some useful relevant information is found in the work of Washington, which is used by the present author in order to obtain first further useful information about the fields \(K_n\) and asymptotic expressions for the roots of \(\phi_n(x)\). Based on this information he then proves a number of approximation properties of the solutions \((x,y)\) of equations (*). All these properties are valid for \(n\geq 2000\). The next step is to prove that equation (*) cannot have solutions with \(xy\neq 0\) if \(n\) is ``large''. This is accomplished by using a sharp lower bound for linear forms in two logarithms due to \textit{M. Laurent, M. Mignotte} and \textit{Yu. Nesterenko} [J. Number Theory 55, No. 2, 285--321 (1995; Zbl 0843.11036)], in order to obtain an upper bound for \(n\) which is of ``medium size'' \(N\), namely \(N=2242936\) in the worst case. Next, all equations (*) in the range \(2000 \leq n\leq N\) are solved. In this step the crucial tool is a lower bound for linear forms in three logarithms, which is obtained from a more general result of \textit{A. Baker} and \textit{G. Wüstholz} [J. Reine Angew. Math. 442, 19--62 (1993; Zbl 0788.11026)]. As a result, inequalities of the form \(\log y\leq c\log n\) are obtained, where, in the worst case, \(c=1.7\times 10^{33}\). This, in combination with a reduction technique, based on condinued fractions and a lemma of C. Heuberger, A. Pethő and R. F. Tichy, shows that there are no nontrivial solutions to equation (*) in the range \(2000\leq n\leq N\). All the necessary computations were made with the aid of Maple. Finally the Thue equations (*) in the range \(1\leq n<2000\) were solved with the black-box ``Thue solver'' of KASH.
scientific article; zbMATH DE number 2083529
scientific article