E011: Heuristic rarity of Mersenne primes

Preview figure for E011

Tags: number-theory, quantitative-exploration, visualization See: Valid Tags.

Highlights

  • Compare observed Mersenne-prime counts to a simple heuristic expectation curve.

  • Visualize cumulative “found vs. expected” as exponent bounds increase.

  • Keep the discussion explicitly empirical (evidence, not proof).

Goal

Mersenne primes are extremely rare. This experiment compares:

  • the observed count of Mersenne primes among prime exponents \(p \le P\)

  • a lightweight heuristic curve that grows slowly with \(P\)

The aim is intuition-building and trend visualization, not a rigorous model.

Background (quick refresher)

Research question

For increasing prime exponent bounds \(P\):

  • how does the cumulative count of “passes” from your scan grow?

  • how does it compare to a simple expectation curve of the form:

    \[ E(P) \approx \sum_{p \le P,\ p\ \mathrm{prime}} \frac{1}{p\ln 2} \]

Why this qualifies as a mathematical experiment

  • Finite procedure: run a finite scan and compute a finite expected sum.

  • Observable(s): cumulative count of passes vs. cumulative expected value.

  • Parameter space: vary \(P\) (and optionally cap runtime per test).

  • Outcome: plots that show agreement/divergence and suggest questions for deeper study.

  • Failure modes: small ranges are noisy; the experiment should clearly label bounds.

Experiment design

Computation

  • From E008 (or a fixed list), get the set of exponents tested and which passed.

  • For each bound \(P\), compute:

    • \(\mathrm{Found}(P)\) = number of passing exponents \(p \le P\)

    • \(E(P)\) = cumulative heuristic sum

  • Plot both curves against \(P\).

Outputs

  • plot: \(P\) vs. Found(P) and E(P)

  • table: selected checkpoints (P, Found, E)

How to run

make run EXP=e011

or:

uv run python -m mathxlab.experiments.e011

Notes / pitfalls

  • The heuristic curve is a comparison tool, not a theorem.

  • Results depend heavily on which exponents were actually tested; record the tested set in the report.

  • A single large Mersenne prime can dominate impressions—keep axes and annotations clear.

Extensions

  • Compare multiple heuristics (still empirical) and see which best matches the observed range.

  • Extend the scan bound and see whether the deviation grows or stabilizes.

Published run snapshot

If this experiment is included in the docs gallery, include the published snapshot (report + params).

Reproduce:

make run EXP=e011

Parameters

  • p_max: 20000

  • max_tests: 900

Summary

  • largest tested p: 6997

  • observed count: 20

  • expected (heuristic): 3.52597

Found Mersenne prime exponents

2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423

Notes

  • The heuristic probability is ~ 1/(p·ln 2) for prime p.

  • This is not a theorem; it is a back-of-the-envelope model for rarity.

params.json (snapshot)
{
  "max_tests": 900,
  "p_max": 20000
}

References

See References.

[Caldwell, 2021, Mersenne Research, 2025]