Lower bound for class numbers of certain real quadratic fields

summary:Let $d$ be a square-free positive integer and $h(d)$ be the class number of the real quadratic field $\mathbb {Q}{(\sqrt {d})}.$ We give an explicit lower bound for $h(n^2+r)$, where $r=1,4$. Ankeny and Chowla proved that if $g>1$ is a natural number and $d=n^{2g}+1$ is a square-free integer, then $g \mid h(d)$ whenever $n>4$. Applying our lower bounds, we show that there does not exist any natural number $n>1$ such that $h(n^{2g}+1)=g$. We also obtain a similar result for the family $\mathbb {Q}(\sqrt {n^{2g}+4})$. As another application, we deduce some criteria for a class group of prime power order to be cyclic