Research

Seven aggregates over 148,933 primes.

Everything on this page was computed from the shipped shards, on the environment named at the bottom, by a script whose hash is printed here. Nothing is quoted from a paper.

dataseta5caea22a57efaac915c…

workflow36f765220749b287cde6…

output5b93e5f26256e16f6290…

environmentPython 3.14.5 · NumPy 2.5.1

Density

148,933 primes below 2,000,000 gives a density of 0.074467. Density falls as the range grows, which is the shape the prime number theorem predicts; the magnitude bands below show the fall directly.

Range	Primes	Density
0–10	4	0.40000
10–100	21	0.23333
100–1,000	143	0.15889
1,000–10,000	1,061	0.11789
10,000–100,000	8,363	0.09292
100,000–1,000,000	68,906	0.07656
1,000,000–2,000,000	70,435	0.07043

The counts in that table sum to the dataset total, and each row is the difference between two consecutive values of π(10ᵏ). They are stored, not re-derived: the band boundaries come from the decimal_digits column that ships inside the shards.

Gaps

Across 148,932 consecutive-prime gaps the mean is 13.429 and the largest is 132. That maximum is the known maximal gap below two million, which is one of the cheapest ways to check this dataset against the literature.

median gap: 10
90th percentile: 28
99th percentile: 52
maximum: 132
distinct gap values: 61

Most frequent gaps

gap 214,871

gap 414,741

gap 624,531

gap 810,280

gap 1012,967

gap 1215,141

gap 147,941

gap 189,559

Gap 6 is the most common by a wide margin — the jumping champion for this range. Gap 2 occurs exactly as often as there are twin pairs, which is a consistency check the two independent columns have to pass.

Residues

Every prime above 5 is coprime to 30, so it must land in one of eight residue classes. It lands in them almost equally:

Primes by residue class mod 30

n ≡ 118,540

n ≡ 718,625

n ≡ 1118,648

n ≡ 1318,654

n ≡ 1718,642

n ≡ 1918,593

n ≡ 2318,601

n ≡ 2918,627

The eight counts span only 114. The dataset also stores residues mod 6 and mod 210, and a derived wheel30 class — the column that generation 2 was proposed and reviewed in order to add.

Families

Four pair families are encoded as bit flags, each set on the larger member of the pair so that appending a later range never rewrites an earlier shard.

Prime family counts

twin prime14,871

cousin prime14,742

sexy prime29,419

safe prime7,746

Cousin pairs exceed gap-4 occurrences by exactly one. The extra pair is (3, 7): 7 − 4 = 3 is prime, but 5 sits between them, so it is a cousin pair that is not a gap of 4. It is the only such case in the range.

Replay

One command, no network, no index required:

shell

python workflows/stable_baseline.py --project-root . \
    --output results_v10/stable_baseline_output.json

The script verifies the stable manifest against its own hash before reading anything, refuses to run if the shard or prime count disagrees with the manifest, and fails loudly if the family-flag column carries a bit it cannot name. Its output is canonicalised and self-hashed the same way the manifest is, so the value below is reproducible byte for byte.

workflow: stable_baseline.py
workflow sha256: 36f765220749b287cde604bbff48b2c1b31e1c212855afeac3c10498bfd22569
dataset manifest: a5caea22a57efaac915c00dd92c655b1126e0b6d9b2b93790e48bc167733e0d1
output sha256: 5b93e5f26256e16f6290cea957df45aaf78de672ca0e80bf17ef2419b4d50ac3
environment: Python 3.14.5, NumPy 2.5.1, Windows

Known limits

The range is fixed at two million. Every figure here is a statement about that range and nothing beyond it.
Pair families are counted at the larger member, so a pair straddling the upper bound is not counted.
The shipped prime-distribution-baseline workflow runs through the data-lake query layer and cannot be replayed from the released package, because the lake index is not part of it. That gap is documented on the workflows page; this study exists to close it.
No result here is conditioned on an externally supplied integer, and no combination of them narrows a factor search.