Lower bound for class number and a proof of Chowla and Yokoi's conjecture

Mohit Mishra

Published 2019-06-24Version 1

Let $d$ be a square-free positive integer and $h(d)$ the class number of the real quadratic field $\mathbb{Q}{(\sqrt{d})}.$ In this paper we give an explicit lower bound for $h(n^2+r)$, where $r=1,4$, and also establish an equivalent criteria to attain this lower bound in terms of special value of Dedekind zeta function. Our bounds enables us to provide a new proof of a well-known conjecture of Chowla and that of Yokoi. Also applying our results, we obtain some criteria for class group of prime power order to be cyclic.