Abstract
Let d be a square-free positive integer and h(d) be the class number of the real quadratic field \(\mathbb{Q}\left({\sqrt d} \right)\). We give an explicit lower bound for h(n2 + r), where r = 1, 4. Ankeny and Chowla proved that if g > 1 is a natural number and d = n2g +1 is a square-free integer, then g ∣ h(d) whenever n > 4. Applying our lower bounds, we show that there does not exist any natural number n > 1 such that h(n2g + 1) = g. We also obtain a similar result for the family \(\mathbb{Q}\left({\sqrt {{n^{2g}} + 4}} \right)\). As another application, we deduce some criteria for a class group of prime power order to be cyclic.
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The author is grateful to the anonymous referee for his/her valuable suggestions and remarks which improved the exposition of the paper.
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Mishra, M. Lower bound for class numbers of certain real quadratic fields. Czech Math J 73, 1–14 (2023). https://doi.org/10.21136/CMJ.2022.0264-21
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DOI: https://doi.org/10.21136/CMJ.2022.0264-21