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Mathematics of Computation

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On the computation of all imaginary quadratic fields of class number one

Authors: Juergen M. Cherubini and Rolf V. Wallisser
Journal: Math. Comp. 49 (1987), 295-299
MSC: Primary 11R29; Secondary 11R11
MathSciNet review: 890271
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Abstract: Let d be the discriminant of an imaginary quadratic field with class number one. If $ d \leqslant - {10^4}$ it is easy to show, using an idea from Stark, that $ h(12d) \leqslant 2\sqrt {\vert d\vert} $, $ h(24d) \leqslant 2\sqrt {\vert d\vert} $ and $ \vert h(24d)\ln (5 + 2\sqrt 6 ) - 2h(12d)\ln (2 + \sqrt 3 )\vert < 50\exp ( - \pi /24 \cdot \sqrt {\vert d\vert} )$. This linear form is estimated for large $ \vert d\vert$ from below with the aid of the quantitative version of Schneider's $ {\alpha ^\beta }$-theorem by Mignotte and Waldschmidt. In the "medium large" region $ 2 \cdot {10^4} \leqslant \vert d\vert \leqslant {10^{34}}$ it is shown by computing the beginning of the continued fraction of $ \ln (5 + 2\sqrt 6 )/\ln (2 + \sqrt 3 )$ that the above relations cannot hold.

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Article copyright: © Copyright 1987 American Mathematical Society