license: cc-by-4.0
task_categories:
- tabular-classification
tags:
- mathematics
- number-theory
- continued-fractions
- zaremba-conjecture
- gpu-computation
- hausdorff-dimension
size_categories:
- 1K<n<10K
pretty_name: 'Zaremba Density: GPU-Computed Exception Sets and Density Data'
Zaremba Density: GPU-Computed Exception Sets and Density Data
Canonical repository for all Zaremba density sweep data. GPU-computed density measurements for Zaremba's conjecture across 100+ digit set / range combinations, computed on an 8x NVIDIA B200 cluster (1.4 TB VRAM).
Part of the bigcompute.science project.
Note: Computational proof framework data (transfer operator, spectral gaps, representation counts) lives in cahlen/zaremba-conjecture-data.
What is Zaremba's Conjecture?
For every positive integer d, there exists a coprime a/d whose continued fraction has all partial quotients bounded by 5. The density of a digit set A at range N is the fraction of integers d in [1, N] that have at least one coprime a/d with all partial quotients in A.
Dataset Contents
zaremba_density_all_results.csv
Summary of all completed experiments: digit set, range, density, uncovered count, runtime.
density_sweep_1e6.csv / density_all_subsets_n10_1e6.csv
Complete density sweep of all 1,023 non-empty subsets of {1,...,10} at N=10^6.
logs/
All raw GPU log files with full output including uncovered denominator lists for closed exception sets. Covers ranges from 10^6 through 10^14.
results/
Analysis JSON and harvested summaries.
Key Results
Closed Exception Sets (verified to 10^11)
| Digit Set | Exceptions | Verified To |
|---|---|---|
| {1,2,3} | 27 | 10^9 (10^11 running) |
| {1,2,4} | 64 | 10^10 (10^12 running) |
| {1,2,5} | 374 | 10^11 |
| {1,2,6} | 1,834 | 10^11 |
| {1,2,7} | 7,178 | 10^11 |
Open (Growing) Exception Sets at 10^11
| Digit Set | Exceptions |
|---|---|
| {1,2,8} | 23,590 |
| {1,2,9} | 77,109 |
| {1,2,10} | 228,514 |
{1,2} Logarithmic Convergence
| Range | Density |
|---|---|
| 10^6 | 61.28% |
| 10^9 | 72.06% |
| 10^10 | 76.55% |
| 10^11 | 80.75% |
| 10^12 | 84.58% |
Fits: density ~ 31.5 + 4.47 * log10(N). Predicts 100% at ~10^15.
Hardware
- 8x NVIDIA B200 (180 GB each, ~1.4 TB total VRAM)
- Custom CUDA kernels with persistent threads, prefix-sorted work distribution
Reproduction
nvcc -O3 -arch=sm_90 -o zaremba_density_gpu zaremba_density_gpu.cu -lm
./zaremba_density_gpu 100000000000 1,2,3
Source code: github.com/cahlen/idontknow/tree/main/scripts/experiments/zaremba-density
Related Datasets
- cahlen/zaremba-conjecture-data — Computational proof framework: transfer operator, spectral gaps, representation counts
- cahlen/hausdorff-dimension-spectrum — dim_H(E_A) for all 2^20-1 subsets
Source
- Code: zaremba-density
- Findings: Phase Transition, Exception Hierarchy
- Project: bigcompute.science
Citation
@misc{humphreys2026zaremba_density,
author = {Humphreys, Cahlen and Claude (Anthropic)},
title = {Zaremba Density: GPU-Computed Exception Sets and Density Data},
year = {2026},
publisher = {Hugging Face},
url = {https://huggingface.co/datasets/cahlen/zaremba-density}
}
Human-AI collaborative work (Cahlen Humphreys + Claude). Not independently peer-reviewed. All code and data open for verification. CC BY 4.0.