Narrow Richaud–Degert Quadratic Fields

• Narrow Richaud–Degert type fields are real quadratic fields defined by discriminants of the form m² ± {1,4}, enabling unit-generated maximal orders and exceptionally short continued fractions.
• The classification into two parametric families, Δₙ⁺ = n² − 4 and Δₙ⁻ = n² + 4, provides concrete criteria for squarefree discriminants, explicit fundamental units, and precise class number evaluations.
• These fields underpin advances in number theory by facilitating efficient Pell equation solutions, yielding asymptotic bounds on class numbers, and supporting computational methods in explicit class field theory.

A real quadratic field $K = \mathbb{Q}(\sqrt{D_0})$ possesses rich arithmetic structure, often encoded in the behavior of its continued fractions, class numbers, fundamental units, and the shape of its maximal order $\mathcal{O}_K$. The narrow Richaud–Degert (RD) type is a critical subclass for both theoretical class number bounds and explicit unit-construction, distinguished by specific arithmetic properties of its discriminant and short continued fractions. The theory finds precise classification in the context of unit-generated orders, with direct algebraic closed forms for fundamental solutions and detailed behavior of the class groups.

1. Definition and Characterization

Let $K = \mathbb{Q}(\sqrt{\Delta_0})$, where $\Delta_0 > 1$ is squarefree and is the field's fundamental discriminant (so $\Delta_0 = D_0$ or $4D_0$ depending on $D_0 \equiv 1 \bmod 4$ or not). $K$ is said to be of narrow Richaud–Degert type if the discriminant can be written as

$$\Delta_0 = D_0 = m^2 + r, \quad \text{with } r \in \{\pm 1, \pm 4\},$$

and $D_0$ is squarefree. Equivalently, $K$ is of narrow RD type if its maximal order $\mathcal{O}_K$ is unit-generated, or there is a unique index-2 unit-generated over-order. The continued fraction expansion of $\sqrt{D}$ for such $D$ is “exceptionally short” (Kopp et al., 12 Dec 2025, Pletser, 2015).

Analytically, a positive integer $D$ is narrow RD if $D = f(u)^2 \pm 2^\alpha n$ for integer polynomial $f(u) > 0$, integers $\alpha \geq 0$, $n > 0$, and $D$ squarefree, with $f(u)$ satisfying either:

• Divisible subcase: $f(u) \equiv 0 \ (\mathrm{mod} \ 2^{\alpha-1} n)$,
• Half-shifted subcase: $$f(u) \equiv 2^{\alpha-2} n \ (\mathrm{mod} \ 2^{\alpha-1} n)$$ (Pletser, 2015).

2. Parametrization of Narrow Richaud–Degert Fields

Every real quadratic unit-generated order falls precisely into one of two one-parameter families: $\Delta_n^+ = n^2 - 4, \qquad n \geq 3, \qquad \text{(norm %%%%22%%%% fundamental unit)}$

$\Delta_n^- = n^2 + 4, \qquad n \geq 1, \qquad \text{(norm %%%%23%%%% fundamental unit)}$

where $n$ is chosen so that $\Delta_n^\pm$ is squarefree or $4 \cdot$ (squarefree) when $\Delta_n^- \equiv 0 \bmod 4$.

The maximal order cases arise as follows:

• $\Delta_0 = m^2 - 4$ with odd $m$: $n = m$ odd, $f = 1$.
• $\Delta_0 = m^2 - 1$ with even $m$: $n = 2m \equiv 0 \bmod 4$, $f = 1$.
• $\Delta_0 = m^2 + 1$ with even $m$: $\Delta = 4\Delta_0 = (2m)^2 + 4$, $n = 2m \equiv 0 \bmod 4$, $f = 2$.
• $\Delta_0 = m^2 + 1$ with odd $m$: $\Delta_0 = (2m)^2 + 4$, $n = 2m \equiv 2 \bmod 4$, $f = 1$.
• $\Delta_0 = m^2 + 4$ with odd $m$: $n = m$ odd, $f = 1$.

Thus, narrow RD-type fields are parametrized by $n$ in these families with suitable squarefree conditions (Kopp et al., 12 Dec 2025).

3. Continued-Fraction Expansions and Fundamental Units

For $D$ of narrow RD form, the continued-fraction expansion of $\sqrt{D}$ is very short (period two, four, eight, ten, or twelve, according to the congruence), facilitating closed-form solutions to the Pell equation $X^2 - DY^2 = 1$.

Divisible Subcase ($f \equiv 0 \bmod m$):

• $D = f^2 + 2^\alpha n$, period $r = 1$:

$\sqrt{f^2 + 2^\alpha n} = [f;\ f/m,\ 2f]$

• Fundamental solution:

$$X_1 = \frac{f^2}{2^{\alpha-1} n} + 1,\quad Y_1 = \frac{f}{2^{\alpha-1} n}$$

• $D = f^2 - 2^\alpha n$, period $r = 3$:

$$\sqrt{f^2 - 2^\alpha n} = [f-1; 1,\ f/m-2,\ 1,\ 2(f-1)]$$

• Fundamental solution:

$$X_1 = \frac{f^2}{2^{\alpha-1} n} - 1,\quad Y_1 = \frac{f}{2^{\alpha-1} n}$$

Half-shifted Subcase ($f \equiv m/2 \bmod m$):

• If $f$ is even: period $r = 7$, fundamental solution for $D = f^2 + 2^\alpha n$:

$$X_1 = \frac{f^2 (f^2 + 2^\alpha n) + 1}{2^{2\alpha-3} n^2},\quad Y_1 = \frac{f(f^2 + 2^{\alpha-1} n)}{2^{2\alpha-3} n^2}$$

• If $f$ is odd: period $r = 9$ or $r = 11$, fundamental solution for $D = f^2 + 2^\alpha n$:

$$X_1 = \frac{f^2 (f^2 + 3 \cdot 2^{\alpha-2} n)^2 + 1}{2^{3\alpha-5} n^3},\quad Y_1 = \frac{f (f^2 + 2^{\alpha-2} n)(f^2 + 3\cdot 2^{\alpha-2} n)}{2^{3\alpha-5} n^3}$$

The minimal polynomials satisfied by fundamental units are $(\varepsilon)^2 - n \varepsilon + 1 = 0$ or $(\varepsilon)^2 - n \varepsilon - 1 = 0$.

4. Classification Theorems: Class Numbers and Torsion

4.1 Asymptotic Brauer–Siegel Bound

For the two families $\Delta_n^\pm$ of unit-generated orders,

$$\log |\mathrm{Cl}(\Delta_n^\pm)| \sim \frac{1}{2} \log \Delta_n^\pm \quad \text{as } |\Delta| \to \infty,$$

establishing that the class numbers of these quadratic orders exhibit rapid growth, governed by a Siegel–Hua formula (Kopp et al., 12 Dec 2025).

4.2 Finiteness of Class Number One and 2-Torsion

The only maximal orders of narrow RD type with class number one are:

• For $\Delta_n^+ = n^2 - 4$, $n \in \{0,1,3,4,5,9,21\}$, yielding $\Delta \in \{-4,-3,5,12,21,77,437\}$.
• For $\Delta_n^- = n^2 + 4$, $n \in \{1,2,3,5,7,13,17\}$, yielding $\Delta \in \{5,8,13,29,53,173,293\}$ (Kopp et al., 12 Dec 2025).

There are only finitely many unit-generated real quadratic orders whose wide class group is 2-torsion; equivalently, the same holds for the narrow class group (orders with one class per genus).

A principal-genus argument bounds $|\mathrm{Cl}(\Delta)| \leq 2^{\mu(\Delta) - 1}$, where $\mu(\Delta) = O(\log \Delta / \log \log \Delta)$, implying $\log |\mathrm{Cl}| = o(\log \Delta)$. For $\Delta_n^\pm$, $$\log |\mathrm{Cl}| \sim \frac{1}{2} \log \Delta \sim \log n$$, excluding the possibility of infinite torsion (Kopp et al., 12 Dec 2025).

5. Numerical Lists of Fields and Class Group Structure

Tables presented in (Kopp et al., 12 Dec 2025) give exhaustive lists up to $\Delta \leq 10^{10}$ for both families:

$n$ for $\Delta_n^+$ (class number 1)	$\Delta = n^2 - 4$	$n$ for $\Delta_n^-$ (class number 1)	$\Delta = n^2 + 4$
0, 1, 3, 4, 5, 6, 7, 9, 11, 21	-4, -3, 5, 12, 21, 32, 45, 77, 117, 437	1, 2, 3, 4, 5, 7, 8, 11, 13, 17	5, 8, 13, 20, 29, 53, 68, 125, 173, 293

The tables also enumerate discriminants whose class group is 2-torsion, both for wide and narrow categories.

6. Classical Cases and General Solutions to the Pell Equation

Classical small cases are recovered as specializations:

• $\alpha = 0$ gives $D = f^2 \pm 1$, recovering $(X_1, Y_1) = (f^2 + 1, f)$ for $D = f^2 + 1$, and $(f^2 - 1, f)$ for $D = f^2 - 1$.
• $\alpha = 1$, $n = 1$ gives $D = f^2 \pm 2$, again producing classical Richaud–Degert forms (Pletser, 2015).

The continued-fraction algorithm yields closed-form expressions for the fundamental solutions $(X_1, Y_1)$ for arbitrary polynomial $f(u)$, facilitating investigation of the unit structure and class numbers for quadratic fields of narrow RD type.

7. Broader Arithmetic Significance

The exceptional periodic structure of the continued fractions for $\sqrt{D}$ in narrow RD cases underpins rapid, explicit characterization of unit groups, regulator growth, and genus theory distinctions. The Siegel–Hua bound demonstrates that the class number escalates rapidly, explaining the finiteness of fields with trivial or exponent-2 class group in these families. Explicit formulas and classification, as in (Kopp et al., 12 Dec 2025) and (Pletser, 2015), facilitate further progress in analytic number theory and Diophantine equations.

A plausible implication is that the structural restrictions in narrow RD fields—manifested by short periodic continued fractions and explicit unit-generation—enable rigorous genus/class group calculations and complete computational lists up to large discriminants, which is crucial for effective explicit class field theory and computational number theory.