# E113: First prime in each residue class

```{figure} ../_static/experiments/e113_hero.png
:alt: E113 hero
:class: experiment-hero

E113: First prime in each residue class
```

**Tags:** `number-theory`, `prime-races`, `visualization`, `aps`, `primes`

## Highlights

- Focused numeric experiment with a small set of figures.
- Parameters saved to `params.json` for reproducibility.
- Defaults are chosen, so the experiment remains feasible for the CI “slow” suite.

## What is computed

- For each reduced residue a (mod q), find the smallest prime p≡a (mod q) and visualize results.

## Notes

- This page summarizes the intent; see the generated `report.md` in `out/` for concrete outputs.

## Published run snapshot

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

```{include} ../reports/e113.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e113.json
:language: json
```

:::

## References

- See `docs/background/primes-in-arithmetic-progressions.md` and `docs/background/prime-number-races.md`.

## Related experiments

- {doc}`e112` (E112: Prime race: π(x;q,a) − π(x;q,b))
- {doc}`e023` (Residue class distribution mod q)
- {doc}`e070` (Primes in residue classes: pi(x; q, a).)
- {doc}`e081` (Prime race sign changes: first crossings table.)
- {doc}`e072` (Prime race mod 4: pi(x;4,3) vs. pi(x;4,1).)