Infinite families of pairs of real quadratic fields whose class numbers are divisible by a given integer

Source abstract

For any integers $N \ge 2$ and $m \ge 1$, we prove that there exist infinitely many pairs of real quadratic fields \(\QQ(\sqrt{D}), \QQ(\sqrt{D + m})\), with \(D \in \ZZ\) and $D > 0$, such that the class numbers of both fields are divisible by $N$. Write $N = 2^{e}n$ with $n$ odd. If $n > 1$, then the class groups of both fields in each such pair contain an element of order $n$.