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

TL;DR

This paper constructs an infinite family of imaginary quadratic fields with class numbers divisible by a given odd prime, confirming a special case of Iizuka's conjecture.

Contribution

It provides a method to generate infinite pairs of imaginary quadratic fields with class numbers divisible by any odd prime, settling a specific case of Iizuka's conjecture.

Findings

Constructed infinite families of quadratic fields with class numbers divisible by p

Confirmed Iizuka's conjecture for n=1 and p>2

Established divisibility properties for class numbers in these fields

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 $p >2.$