Algebraic number fields with large class number

We prove that “almost all” real quadratic fields of a given type have a large ideal class number. For example, the number of ideal classes of the fields $\mathbf Q\bigl(\sqrt{m(m+1)(m+2)(m+3)}\,\bigr)$, where $\mathbf Q$ is the field of rational numbers, grows unbounded with $m$, as $m$ ranges through all natural numbers, except for a very sparse sequence. An analogous fact is established for the fields of Ankeny–Brauer–Chowla [5].