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

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 $h(12d) \leqslant 2\sqrt {|d|}$, $h(24d) \leqslant 2\sqrt {|d|}$ and $|h(24d)\ln (5 + 2\sqrt 6 ) - 2h(12d)\ln (2 + \sqrt 3 )| < 50\exp ( - \pi /24 \cdot \sqrt {|d|} )$. This linear form is estimated for large $|d|$ from below with the aid of the quantitative version of Schneider’s ${\alpha ^\beta }$-theorem by Mignotte and Waldschmidt. In the "medium large" region $2 \cdot {10^4} \leqslant |d| \leqslant {10^{34}}$ 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.