Consecutive real quadratic fields with large class numbers

TL;DR

This paper proves that for any positive integer k, there are many consecutive real quadratic fields with large class numbers, approaching the maximum possible, within a certain range.

Contribution

It establishes the existence of numerous consecutive real quadratic fields with near-maximal class numbers, extending understanding of class number distribution.

Findings

At least x^{1/2 - o(1)} such integers d are found up to x.

Consecutive quadratic fields in this range have large class numbers.

The result applies uniformly for any fixed positive integer k.

Abstract

For a given positive integer $k$, we prove that there are at least $x^{1/2-o(1)}$ integers $d\leq x$ such that the real quadratic fields $\mathbb Q(\sqrt{d+1}),\dots,\mathbb Q(\sqrt{d+k})$ have class numbers essentially as large as possible.