Class Numbers of Real Quadratic Fields: Extending Tables to $10^{13}$

Abstract

We compute class numbers $h(d)$ of real quadratic fields $\mathbb{Q}(\sqrt{d})$ for all fundamental discriminants $d$ from $10^{11}$ to $10^{13}$ — a 100× extension of existing systematic tables. Each discriminant is handled by an independent CUDA thread using the analytic class number formula combined with continued-fraction computation of the regulator. The resulting dataset tests the Cohen-Lenstra heuristics at unprecedented scale, specifically the prediction that $h(d) = 1$ occurs with probability $\approx 75.446\%$ for real quadratic fields.

Background

Class Numbers

The class number $h(K)$ of a number field $K$ measures the failure of unique factorization in its ring of integers. For the real quadratic field $\mathbb{Q}(\sqrt{d})$:

• $h(d) = 1$ means the ring $\mathbb{Z}[\frac{1+\sqrt{d}}{2}]$ (or $\mathbb{Z}[\sqrt{d}]$) has unique factorization
• $h(d) > 1$ means unique factorization fails, and the class group has non-trivial structure

The Analytic Class Number Formula

For a fundamental discriminant $d > 0$:

$h(d) \cdot R(d) = \frac{\sqrt{d}}{2} \cdot L(1, \chi_d)$

where:

• $R(d) = \log \varepsilon_d$ is the regulator (logarithm of the fundamental unit)
• $L(1, \chi_d) = \sum_{n=1}^{\infty} \frac{\chi_d(n)}{n}$ is the Dirichlet $L$-function
• $\chi_d(n) = \left(\frac{d}{n}\right)$ is the Kronecker symbol

The regulator is computed from the continued fraction expansion of $\sqrt{d}$: the period of the CF gives the fundamental unit $\varepsilon_d$, and $R(d) = \log \varepsilon_d$.

The Cohen-Lenstra Heuristics

Cohen and Lenstra (1984) proposed a remarkable probabilistic model for class groups. For real quadratic fields, they predict:

$\text{Prob}(h(d) = 1) = \prod_{p \text{ prime}} \left(1 - \frac{1}{p}\right)\left(1 - \frac{1}{p^2}\right) \approx 0.75446$

More generally, they predict the distribution of the odd part of the class group: an abelian group $G$ occurs with probability proportional to $1/|\text{Aut}(G)|$.

These heuristics have been tested computationally up to $d \sim 10^{11}$ and agree beautifully, but more data at larger discriminants is needed to:

1. Test whether the convergence continues
2. Look for deviations that might reveal deeper structure
3. Understand the distribution of large class numbers

Current Frontier

Authors	Year	Range	Key Result
Jacobson, Ramachandran, Williams	2006-2015	$d \leq 10^{11}$	Systematic class numbers via BSGS
Watkins	2004	Imaginary quadratic, $d \leq 10^{10}$	Complete tables
Belabas, Cohen	Various	Cubic fields, $d \leq 10^7$	Limited by algorithm complexity

No systematic GPU-accelerated computation exists. The BSGS algorithm is embarrassingly parallel — each discriminant is independent — making it ideal for GPU acceleration.

Hardware

Component	Specification
Node	NVIDIA DGX B200
GPUs	8× NVIDIA B200 (183 GB VRAM each)
Total VRAM	1.43 TB
GPU Interconnect	NVLink 5 (NV18), full mesh
CPUs	2× Intel Xeon Platinum 8570
System RAM	2 TB DDR5

Method

Per-Discriminant Computation

Each CUDA thread handles one discriminant $d$:

1. Check fundamentality: Verify $d$ is a fundamental discriminant (squarefree conditions on $d \bmod 4$)
2. Compute regulator: Run the continued fraction expansion of $\sqrt{d}$ until the period is found, extract the fundamental unit $\varepsilon_d$, compute $R(d) = \log \varepsilon_d$
3. Approximate $L(1, \chi_d)$: Sum the Dirichlet series $\sum_{n \leq N} \chi_d(n)/n$ with $N = O(\sqrt{d})$ terms
4. Extract $h(d)$: Round $\frac{\sqrt{d} \cdot L(1,\chi_d)}{2 R(d)}$ to the nearest integer

Kronecker Symbol

The Kronecker symbol $\left(\frac{d}{n}\right)$ is computed via quadratic reciprocity (Jacobi symbol algorithm) in $O(\log n)$ steps — fast enough for the per-thread L-function summation.

Parallelization

8 GPUs, each handling a contiguous range of discriminants from $10^{11}$ to $10^{13}$.

Results

PENDING — experiment not yet run.

Class Number Distribution

$h$	Count	Fraction	Cohen-Lenstra prediction
1	PENDING	PENDING	$\approx 75.446\%$
2	PENDING	PENDING	PENDING
3	PENDING	PENDING	PENDING
4	PENDING	PENDING	PENDING
$\geq 5$	PENDING	PENDING	PENDING

Convergence of Cohen-Lenstra

Range	Observed $h=1$ fraction	C-L prediction	Ratio
$d \leq 10^{11}$	~75.4% (known)	75.446%	~1.000
$10^{11} < d \leq 10^{12}$	PENDING	75.446%	PENDING
$10^{12} < d \leq 10^{13}$	PENDING	75.446%	PENDING

Reproducibility

git clone https://github.com/cahlen/idontknow
cd idontknow
nvcc -O3 -arch=sm_100a -o class_number_rqf scripts/experiments/class-numbers/class_number_rqf.cu -lm

# Quick test: d = 5 to 10000
./class_number_rqf 5 10000

# Full run: extend to 10^13
./scripts/experiments/class-numbers/run.sh

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Computed on NVIDIA DGX B200. Code: github.com/cahlen/idontknow.