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Datasets:
cahlen
/
zaremba-density

Tasks:
Tabular Classification
Size:
1K<n<10K
Tags:
mathematics
number-theory
continued-fractions
zaremba-conjecture
gpu-computation
hausdorff-dimension
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mask
int64
digits
string
cardinality
int64
density
float64
uncovered
int64
1
{1}
1
0.0029
999,971
2
{2}
1
0.0016
999,984
3
{1,2}
2
57.982
420,180
4
{3}
1
0.0012
999,988
5
{1,3}
2
23.0092
769,908
6
{2,3}
2
0.9238
990,762
7
{1,2,3}
3
99.9973
27
8
{4}
1
0.001
999,990
9
{1,4}
2
7.9261
920,739
10
{2,4}
2
0.3892
996,108
11
{1,2,4}
3
99.9936
64
12
{3,4}
2
0.1169
998,831
13
{1,3,4}
3
99.4333
5,667
14
{2,3,4}
3
24.6132
753,868
15
{1,2,3,4}
4
99.9998
2
16
{5}
1
0.0009
999,991
17
{1,5}
2
3.7524
962,476
18
{2,5}
2
0.2153
997,847
19
{1,2,5}
3
99.9627
373
20
{3,5}
2
0.0737
999,263
21
{1,3,5}
3
98.7217
12,783
22
{2,3,5}
3
16.0505
839,495
23
{1,2,3,5}
4
99.9997
3
24
{4,5}
2
0.0421
999,579
25
{1,4,5}
3
77.6313
223,687
26
{2,4,5}
3
7.7765
922,235
27
{1,2,4,5}
4
99.9997
3
28
{3,4,5}
3
2.2568
977,432
29
{1,3,4,5}
4
99.9964
36
30
{2,3,4,5}
4
84.0646
159,354
31
{1,2,3,4,5}
5
100
0
32
{6}
1
0.0008
999,992
33
{1,6}
2
2.1705
978,295
34
{2,6}
2
0.1358
998,642
35
{1,2,6}
3
99.828
1,720
36
{3,6}
2
0.057
999,430
37
{1,3,6}
3
95.653
43,470
38
{2,3,6}
3
11.6778
883,222
39
{1,2,3,6}
4
99.9997
3
40
{4,6}
2
0.0306
999,694
41
{1,4,6}
3
74.8903
251,097
42
{2,4,6}
3
5.3962
946,038
43
{1,2,4,6}
4
99.9998
2
44
{3,4,6}
3
1.5727
984,273
45
{1,3,4,6}
4
99.9964
36
46
{2,3,4,6}
4
74.9406
250,594
47
{1,2,3,4,6}
5
99.9999
1
48
{5,6}
2
0.0199
999,801
49
{1,5,6}
3
43.4333
565,667
50
{2,5,6}
3
3.6154
963,846
51
{1,2,5,6}
4
99.9976
24
52
{3,5,6}
3
1.0086
989,914
53
{1,3,5,6}
4
99.9929
71
54
{2,3,5,6}
4
60.8832
391,168
55
{1,2,3,5,6}
5
99.9999
1
56
{4,5,6}
3
0.521
994,790
57
{1,4,5,6}
4
99.5331
4,669
58
{2,4,5,6}
4
37.6467
623,533
59
{1,2,4,5,6}
5
100
0
60
{3,4,5,6}
4
12.793
872,070
61
{1,3,4,5,6}
5
99.9995
5
62
{2,3,4,5,6}
5
99.1446
8,554
63
{1,2,3,4,5,6}
6
100
0
64
{7}
1
0.0008
999,992
65
{1,7}
2
1.4305
985,695
66
{2,7}
2
0.1157
998,843
67
{1,2,7}
3
99.4612
5,388
68
{3,7}
2
0.0437
999,563
69
{1,3,7}
3
89.518
104,820
70
{2,3,7}
3
9.0632
909,368
71
{1,2,3,7}
4
99.9992
8
72
{4,7}
2
0.0242
999,758
73
{1,4,7}
3
64.0974
359,026
74
{2,4,7}
3
4.2324
957,676
75
{1,2,4,7}
4
99.9995
5
76
{3,4,7}
3
1.2401
987,599
77
{1,3,4,7}
4
99.9931
69
78
{2,3,4,7}
4
68.468
315,320
79
{1,2,3,4,7}
5
100
0
80
{5,7}
2
0.0171
999,829
81
{1,5,7}
3
43.4232
565,768
82
{2,5,7}
3
2.6206
973,794
83
{1,2,5,7}
4
99.999
10
84
{3,5,7}
3
0.7887
992,113
85
{1,3,5,7}
4
99.9933
67
86
{2,3,5,7}
4
53.3554
466,446
87
{1,2,3,5,7}
5
100
0
88
{4,5,7}
3
0.433
995,670
89
{1,4,5,7}
4
99.5327
4,673
90
{2,4,5,7}
4
31.6389
683,611
91
{1,2,4,5,7}
5
100
0
92
{3,4,5,7}
4
10.4695
895,305
93
{1,3,4,5,7}
5
99.9996
4
94
{2,3,4,5,7}
5
98.4429
15,571
95
{1,2,3,4,5,7}
6
100
0
96
{6,7}
2
0.015
999,850
97
{1,6,7}
3
23.9803
760,197
98
{2,6,7}
3
1.9576
980,424
99
{1,2,6,7}
4
99.9915
85
100
{3,6,7}
3
0.5819
994,181
End of preview.

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

Source

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.
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