On the application of Skolem's p-adic method to the solution of Thue equations
The authors show by rather ingenious methods that the only integer solutions of the diophantine equation \[ (*)\quad x^ 4-4x^ 2y^ 2+y^ 4=-47 \] are \((x,y)=(\pm 2,\pm 3)\) and \((\pm 3,\pm 2)\). This seems to be the first case where a totally real binary quartic has been successfully attempted by methods of algebraic number theory. The authors use a trick of Ljunggren - working in a number field M of degree 8 over \({\mathbb{Q}}\)- to reduce (*) to a p-adic system to which Skolem's method is successfully applied with prime \(p=71\). The numerical hardest job is to find all ``exceptable'' units in M for which task they give a powerful reduction algorithm [see also the second author and \textit{B. M. M. de Weger}, ibid. 31, No.1, 99-132 (1989; Zbl 0657.10014)].
scientific article; zbMATH DE number 4103171
scientific article