de Weger, B. M. M. A hyperelliptic diophantine equation related to imaginary quadratic number fields with class number 2. (English) Zbl 0738.11032 J. Reine Angew. Math. 427, 137-156 (1992). It is proved that \((X,Y)=(2,\pm3),(5,\pm36)\) are the only solutions of the diophantine equation \(X^ 6+10X^ 3-27=13Y^ 2\), as was conjectured by Antoniadis. This yields a new proof of the fact that the only fields \(\mathbb{Q}(\sqrt{-13p})\) with \((p,6)=1\) having class number two are those with \(p=7,31\). The mentioned diophantine equation is reduced to a cubic Thue equation with coefficients in \(\mathbb{Q}(\sqrt{13})\), which is then solved by the theory of linear forms in logarithms and computational diophantine approximation techniques. Some conjectures on related ‘Thue- recurrence equations’ are formulated. Reviewer: B.M.M.de Weger Cited in 1 ReviewCited in 3 Documents MSC: 11D41 Higher degree equations; Fermat’s equation 11R11 Quadratic extensions 11R29 Class numbers, class groups, discriminants 11J86 Linear forms in logarithms; Baker’s method Keywords:imaginary quadratic fields; hyperelliptic diophantine equation; sixth degree diophantine equations; class number two; cubic Thue equation; linear forms in logarithms; Thue-recurrence equations PDFBibTeX XMLCite \textit{B. M. M. de Weger}, J. Reine Angew. Math. 427, 137--156 (1992; Zbl 0738.11032) Full Text: DOI Crelle EuDML