Pethő's cubics (Q2707224)

In this paper the family of cubic Thue equations NEWLINE\[NEWLINEx^3-nx^2y-(n+1)xy^2-y^3=1,NEWLINE\]NEWLINE where \(n\in \mathbb Z\), is completely solved, superseding the former result of the author and \textit{N. Tzanakis} [J. Number Theory 39, 41-49 (1991; Zbl 0734.11025)], where this was done only for \(n>3.67\cdot 10^{32}\). For \(n\not\in \{-5,-4,-1,0,3,4\}\) the above Thue equation has the solutions \((x,y)\in \{(1,0), (0,-1), (1,-1), (-n-1,-1), (1,-n)\}\). By careful estimations and using the recent bounds for linear forms of two (resp. three) logarithms, this is proved for \(|n|>810000\). For the remaining values of \(n\) a lemma à la Baker-Davenport enables the computer to perform the necessary calculations within some hours.