Let $K=\mathbb{Q}(\sqrt{d})$ be a real quadratic fields. S. Chowla conjectured that if $d=4m^2+1$ for $m \in \mathbb{Z}$ then there is exactly 6 real quadratic fields which has class number one. Hideo Yokoi proved this conjecture (almost!). Then the natural question arises: What can be said about similar real quadratic fields $\mathbb{Q}(\sqrt{a^2m^2+1})$ for $a \not = 2$ and $m$ a integers? I am looking for suitable reference for this question? Can someone suggest reference for this question?
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$\begingroup$ Don't you mean $a \not = 2$? $\endgroup$– GlorfindelJul 12, 2016 at 21:08
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1$\begingroup$ Not only $a \ne 2$, but $a = 1$. And he probably googled "New York mining desaster" instead of "Chowla real quadratic". $\endgroup$– Franz LemmermeyerJul 13, 2016 at 4:33
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$\begingroup$ If you like my answer, please accept it officially (so that it turns green). Thanks in advance! $\endgroup$– GH from MOSep 2, 2018 at 7:55
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The conjectures of Chowla and Yokoi were proved by Biró (2003). The state-of-the-art of this problem is contained in the following preprint by Biró and Lapkova (2015).
answered Jul 12, 2016 at 17:16
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1$\begingroup$ The paper of Biró and Lapkova was published as: Acta Arith. 172 (2016), no. 2, 117–131. eudml.org/doc/279595 $\endgroup$ Feb 28, 2021 at 9:33