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 ⩽ − 10 4 d \leqslant - {10^4} it is easy to show, using an idea from Stark, that h ( 12 d ) ⩽ 2 | d | h(12d) \leqslant 2\sqrt {|d|} , h ( 24 d ) ⩽ 2 | d | h(24d) \leqslant 2\sqrt {|d|} and | h ( 24 d ) ln ⁡ ( 5 + 2 6 ) − 2 h ( 12 d ) ln ⁡ ( 2 + 3 ) | > 50 exp ⁡ ( − π / 24 ⋅ | d | ) |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 | |d| from below with the aid of the quantitative version of Schneider’s α β {\alpha ^\beta } -theorem by Mignotte and Waldschmidt. In the "medium large" region 2 ⋅ 10 4 ⩽ | d | ⩽ 10 34 2 \cdot {10^4} \leqslant |d| \leqslant {10^{34}} it is shown by computing the beginning of the continued fraction of ln ⁡ ( 5 + 2 6 ) / ln ⁡ ( 2 + 3 ) \ln (5 + 2\sqrt 6 )/\ln (2 + \sqrt 3 ) that the above relations cannot hold.