Proof of a conjecture of Guy on class numbers

TL;DR

This paper proves a conjecture by Richard Guy relating class numbers of quadratic fields for primes congruent to 3 mod 4, providing a formula for their ratio modulo powers of 2 using continued fractions.

Contribution

It establishes a formula for the ratio of class numbers of quadratic fields modulo 16, connecting class number ratios to continued fraction expansions, thus solving Guy's conjecture.

Findings

The ratio h(p)/h(-p) is congruent to an integer m(p) modulo 16.

The integer m(p) is derived from the negative continued fraction expansion of √p.

The result confirms a specific modular relationship between class numbers for primes p ≡ 3 (mod 4).

Abstract

It is well known that for any prime $p\equiv 3$ (mod $4$), the class numbers of the quadratic fields $\mathbb{Q}(\sqrt{p})$ and $\mathbb{Q}(\sqrt{-p})$, $h(p)$ and $h(-p)$ respectively, are odd. It is natural to ask whether there is a formula for $h(p)/h(-p)$ modulo powers of $2$. We show the formula $h(p) \equiv h(-p) m(p)$ (mod $16$), where $m(p)$ is an integer defined using the "negative" continued fraction expansion of $\sqrt{p}$. Our result solves a conjecture of Richard Guy.