Complete solutions to a family of cubic Diophantine equations (Q911624)

It is proved in this paper that if \(n\geq 1.365\cdot 10^7\), then the Diophantine equation \[ f_n(x,y)=x^3-(n-1)x^2y-(n+2)xy^2-y^3=\pm 1, \tag{1} \] has only the ``trivial'' solutions \((\pm 1,0)\); \((0,\pm 1)\); \((\pm 1,\mp 1)\). The proof follows that standard method, which is used for the derivation of an effective upper bound for the solutions of Thue equations from an A. Baker's type linear form theorem, but incorporates also a new idea. Let \(x,y\in\mathbb Z^2\) be a solution of (1), \(\alpha(n)\) be a zero of \(f_n(x,1)\) and \(\varepsilon(n)\), \(\eta(n)\) be fundamental units of \(\mathbb Z[\alpha (n)]\). Then there exist integers \(u,v\in\mathbb Z\) with \(x-\alpha(n)y=\varepsilon(n)^u\eta (n)^v\). The new idea is that it is possible to show that if \(uv\neq 0\) then \(\min \{| u|,| v| \}>cn \log n\) with a constant \(c>0\).