On a family of cubics
This is one more step towards the ``practical solution'' of Thue equations invented by the second author and \textit{B. M. M. de Weger} [J. Number Theory 31, 99-132 (1989; Zbl 0657.10014)]. After solving single equations, now the method is able to solve families of equations. The first one was the ``simplest'' cubic family \[ X^ 3-(n-1)X^ 2Y- (n+2)XY^ 2-Y^ 3=\pm 1, \] which was completely solved [see \textit{E. Thomas}, ibid 34, 235-250 (1990; Zbl 0697.10011)] and the first author [Verification of a conjecture of Thomas, Preprint, Strasbourg]. The second example, investigated in this paper, is the family of cubic equations \[ (*)\quad f(X,Y):\;X^ 3-nX^ 2Y-(n+1)XY^ 2-Y^ 3=1. \] A new difficulty arose now, because the cubic extension given by one root of \(f(X,1)=0\) isn't Galois any more. Nevertheless, the authors are able to show that the only solutions of (*) for \(n>3.67\times 10^{32}\) are \[ (1,0),\quad (0,-1),\quad (1,-1),\quad (-n-1,-1),\quad (1,-n). \]
scientific article; zbMATH DE number 4214114
scientific article