# E049: Wieferich primes (base 2): scan and quotient visualization

```{figure} ../_static/experiments/e049_hero.png
:width: 80%
:alt: Preview figure for E049
```

```{figure} ../_static/experiments/e049_hero_2.png
:width: 80%
:alt: Preview figure for E049
```

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

## Highlights

- Scans primes up to a bound for the Wieferich condition $2^{p-1}\equiv 1\ (\mathrm{mod}\ p^2)$.
- Recovers the known small hits (1093 and 3511) under default settings.
- Visualizes a quotient-like value that is exactly zero for Wieferich primes.

## Goal

Explore a rare strengthening of Fermat’s congruence and visualize how exceptional Wieferich primes are.

## Background (quick refresher)

- {doc}`../background/wieferich-primes`
- {doc}`../background/prime-numbers`

## Research question

Up to a bound $B$, which primes satisfy the Wieferich condition (base 2), and how does the quotient-like statistic vary across primes?

## Experiment design

- Generate all primes $p\le B$ (excluding $p=2$ for base 2).
- Compute $r = 2^{p-1} \bmod p^2$ and derive $q = (r-1)/p\ (\mathrm{mod}\ p)$.
- Mark primes with $q=0$ as Wieferich hits.
- Plot hit positions and a scatter of $q$ values.

## Reproducibility

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

## Interpreting the results

- Hits at 1093 and 3511 confirm the implementation.
- The quotient scatter is purely exploratory; it’s a compact way to see “how far” a prime is from the Wieferich condition.
- Increasing the bound quickly becomes compute-heavy (but remains feasible for moderate bounds in pure Python).

## Published run snapshot

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

```{include} ../reports/e049.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e049.json
:language: json
```

:::

## References

See {cite:t}`Wieferich1909ZumLetztenFermatschenTheorem`, {cite:t}`OEISFoundationInc2025A001220WieferichPrimes`, {cite:t}`WikipediaContributors2025WieferichPrime`.

## Related experiments

- {doc}`e018` (Miller–Rabin base choice counterexamples)
- {doc}`e048` (Carmichael numbers: Korselt scan + Fermat counterexamples)
- {doc}`e008` (Lucas–Lehmer scan (prime exponents))
- {doc}`e009` (Small-factor scan for Mersenne numbers)
- {doc}`e042` (Repunit primes (small k scan))