# E010: Even perfect numbers from Mersenne primes

```{figure} ../_static/experiments/e010_hero.png
:width: 80%
:alt: Preview figure for E010
```

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

## Highlights

- Generate even perfect numbers via the Euclid–Euler theorem.
- Connect Mersenne prime exponents $p$ to perfect numbers $N = 2^{p-1}(2^p-1)$.
- Visualize growth and verify the defining property $\sigma(N) = 2N$ for sampled cases.

## Goal

Make the Mersenne $\leftrightarrow$ perfect-number connection computationally explicit:

If $M_p = 2^p - 1$ is prime, then:

$$
N_p = 2^{p-1}(2^p-1)
$$

is an **even perfect number**.

## Background (quick refresher)

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

## Research question

Across the Mersenne prime exponents discovered within your scan bounds:

- how fast do the corresponding even perfect numbers grow?
- can we verify perfectness ($\sigma(N)=2N$) efficiently for these cases?

## Why this qualifies as a mathematical experiment

- **Finite procedure:** find a set of Mersenne primes within a finite bound and generate perfect numbers.
- **Observable(s):** size metrics (digits), and validation checks of $\sigma(N)=2N$.
- **Parameter space:** vary the exponent bound and validation depth.
- **Outcome:** concrete examples + growth plots that support intuition.
- **Reproducibility:** exponents tested and successes recorded in artifacts.

## Experiment design

### Computation

- Obtain a list of Mersenne prime exponents $p$ from a scan (or a fixed list for small $p$).
- For each $p$, compute $N_p = 2^{p-1}(2^p-1)$.
- Verify perfectness for these cases:

  $$
  \sigma(N_p) = 2N_p.
  $$

For modest $p$, exact computation is feasible; for larger $p$, report size metrics and skip expensive checks.

### Outputs

- table: $p$, $M_p$ size, $N_p$ size, and validation status
- plot: $p$ vs. digits of $N_p$
- optional: prime-factor structure display for small cases

## How to run

```bash
make run EXP=e010
```

or:

```bash
uv run python -m mathxlab.experiments.e010
```

## Notes / pitfalls

- Don’t attempt divisor-sum sieves for huge $N_p$; validation must be bounded and explicit.
- Clearly separate “constructed from theorem” (conditional on $M_p$ being prime) from “validated by computation”.

## Extensions

- Cross-link to E002 outputs for perfect numbers and compare growth on the same axes.
- Explore which parts of the perfectness check can be done symbolically using known factorization.

## Published run snapshot

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

```{include} ../reports/e010.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e010.json
:language: json
```

:::

## References

See {doc}`../references`.

{cite:p}`Caldwell2000EvenPerfectNumbersPrimePages,WikipediaContributors2025MersennePrime,OEIS2025A000043MersenneExponents`

## Related experiments

- {doc}`e002` (Even Perfect Numbers — Generator and Growth)
- {doc}`e007` (Mersenne growth (bits and digits))
- {doc}`e009` (Small-factor scan for Mersenne numbers)
- {doc}`e011` (Heuristic rarity of Mersenne primes)
- {doc}`e097` (E097: σ(n)/n landscape: deficient, perfect, abundant)