Class number one problem for a family of real quadratic fields. Issue 4 (11th September 2022)

• Record Type: Journal Article
• Authors: Biró, András
• Is Part Of: Mathematika. Volume 68:Issue 4(2022)
• Page Start: 1221
• Page End: 1232
• Publication Date: 2022-09-11
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1112/mtk.12164
• Languages: English
• ISSNs: 0025-5793

Abstract

Abstract: We effectively solve the class number one problem for a certain family Q ( D ) ${\bf Q}(\sqrt D)$ ( D ∈ F $(D\in {\cal F}$ ) of real quadratic fields, where F $ {\cal F}$ is an infinite subset of the set of odd positive fundamental discriminants. The set F ${\cal F}$ contains the Yokoi discriminants n 2 + 4 $n^2+4$, so our result is a generalization of the solution of Yokoi's Conjecture. But this family may contain also infinitely many fields with comparatively larger fundamental units than the fields in the Yokoi family (it may be as large as log 2 D $\log ^2D$ instead of log D $\log D$ ). The proof is also a generalization of the proof of Yokoi's Conjecture.