# E011: Heuristic rarity of Mersenne primes

```{figure} ../_static/experiments/e011_hero.png
:width: 80%
:alt: Preview figure for E011
```

**Tags:** `number-theory`, `quantitative-exploration`, `visualization`
See: {doc}`../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)

- {doc}`../background/mersenne-primes`

## 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

```bash
make run EXP=e011
```

or:

```bash
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).

```{include} ../reports/e011.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

::: {dropdown} params.json (snapshot)
:open:

```{literalinclude} ../params/e011.json
:language: json
```

:::

## References

See {doc}`../references`.

{cite:p}`Caldwell2021MersennePrimesPrimePages,MersenneResearchInc2025KnownMersennePrimesList`

## Related experiments

- {doc}`e007` (Mersenne growth (bits and digits))
- {doc}`e009` (Small-factor scan for Mersenne numbers)
- {doc}`e010` (Even perfect numbers from Mersenne primes)
- {doc}`e027` (Record prime gaps vs. log^2 heuristic)
- {doc}`e029` (Twin primes: observed vs. heuristic)