# E047: Fermat numbers: Pépin test + factor witnesses

```{figure} ../_static/experiments/e047_hero.png
:width: 80%
:alt: Preview figure for E047
```

```{figure} ../_static/experiments/e047_hero_2.png
:width: 80%
:alt: Preview figure for E047
```

**Tags:** `number-theory`, `counterexample-search`, `visualization`, `fermat`
See: {doc}`../tags`.

## Highlights

- Runs Pépin’s test for $F_n = 2^{2^n} + 1$ (definitive for Fermat numbers).
- Finds small, explicit factor witnesses by bounded trial division.
- Visualizes how explosively $F_n$ grows and where the first counterexample appears.

## Goal

Demonstrate (computationally) why the statement “all Fermat numbers are prime” fails, and produce concrete witnesses.

## Background (quick refresher)

- {doc}`../background/fermat-numbers`

## Research question

How quickly do Fermat numbers become composite in practice, and how easy is it to exhibit a *witness* factor for the first failures?

## Experiment design

- Compute $F_n$ for $n\le n_{max}$.
- Apply Pépin’s test (for $n\ge 1$) to classify prime vs. composite.
- If composite, attempt to find a small factor by trial division up to a configurable bound.
- Plot $\log_{10}(F_n)$ and the smallest discovered factor (if any).

## Reproducibility

- `params.json` records the run settings.
- `report.md` summarizes the key findings.
- `figures/*.png` contains the plots for the run.

## Interpreting the results

- If Pépin says “composite”, the number is definitively composite (not a probabilistic claim).
- The factor search is bounded: “no factor found” does not mean “prime”.
- The classic factor for $F_5$ (and often for $F_6$) appears quickly under the default bounds.

## Published run snapshot

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

```{include} ../reports/e047.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e047.json
:language: json
```

:::

## References

See {cite:t}`KrizekLucaSomer2013LecturesOnFermatNumbers`, {cite:t}`WikipediaContributors2025FermatNumber`, {cite:t}`OEISFoundationInc2025A000215FermatNumbers`.

## Related experiments

- {doc}`e012` (Fermat pseudoprimes and Carmichael numbers (counterexamples))
- {doc}`e048` (Carmichael numbers: Korselt scan + Fermat counterexamples)
- {doc}`e009` (Small-factor scan for Mersenne numbers)
- {doc}`e015` (Wilson test infeasibility)
- {doc}`e041` (Fermat numbers: not all prime)