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On the insolubility of a class of diophantine equations and the nontriviality of the class numbers of related real quadratic fields of Richaud-Degert type. (English) Zbl 0591.12005
We establish criteria for the insolubility in integers \((x,y)\) of \(x^ 2-ny^ 2=\pm 4t\) where \(t\) is a positive integer and \({\mathbb{Q}}(\sqrt{n})\) is of Richaud-Degert (R-D) type. These results are then used to establish the nontriviality of the class number of \({\mathbb{Q}}(\sqrt{n})\) for a large class of R-D types. Tables of values for the class numbers and related diophantine equations are also provided.
Immediate consequences of the above results are results in the literature of N. Ankeny, S. Chowla, H. Hasse, H. Takeuchi, S. D. Lang, and H. Yokoi.

MSC:
11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
11D09 Quadratic and bilinear Diophantine equations
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