On families of parametrized Thue equations
There is an extensive literature of parametric Thue equations [see \textit{C. Heuberger, A. Pethő} and \textit{R. F. Tichy} [Complete solution of parametrized Thue equations, Acta Math. Inform. Univ. Ostrav. 6, 93-113 (1998)]. The powerful result of the paper is a breakthrough in the sense that the author deals with a general type of parametric families of Thue equations, determined by several constants. Let \(d_2,\ldots,d_{n-1}\) \((n\geq 4)\) be pairwise distinct integers and \(a\) an integral parameter. Assuming \[ d_2+\ldots+d_{n-1}\neq 0\quad \text{or}\quad d_2\cdots d_{n-1}\neq 0 \] the author proves, that for \(a\geq a_0(n,d_2,\ldots,d_{n-1})\) all solutions of the equation \[ F_a(X,Y)=(X+aY)(X-d_2Y)(X-d_3Y)\cdots(X-d_{n-1}Y)(X-aY)-Y^n=\pm 1 \] are \((1,0),(-a,1),(d_2,1),(d_3,1),\ldots,(d_{n-1},1),(a,1)\) and their negatives. The ideas of the proof will certainly have further applications.
scientific article; zbMATH DE number 1310083