On the $p$-divisibility of class numbers of an infinite family of imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$ and $\mathbb{Q}(\sqrt{d+1}).$

Abstract

For any odd prime $p,$ we construct an infinite family of pairs of imaginary quadratic fields $\mathbb{Q}(\sqrt{d}),\mathbb{Q}(\sqrt{d+1})$ whose class numbers are both divisible by $p.$ One of our theorems settles Iizuka's conjecture for the case $n=1$ and $p>2.$

arXiv comment6 pages

References in corpus (1)

• A note on the Fourier coefficients of a Cohen-Eisenstein series