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A hyperelliptic diophantine equation related to imaginary quadratic number fields with class number 2. (English) Zbl 0738.11032

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

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
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