# E072: Prime race mod 4: pi(x;4,3) vs. pi(x;4,1).

```{figure} ../_static/experiments/e072_hero.png
:width: 80%
:alt: Preview figure for E072
```

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

## Highlights
- Computes prime-race differences π(x;q,a) − π(x;q,b) on a grid of x values.
- Visualizes lead changes and the size of fluctuations as x grows.
- Adds a derived statistic (normalization / θ-variant) to compare behaviors.

## What this experiment does

A classical "prime race" compares how often one residue class leads another in prime counts. The most famous is mod 4:

The implementation focuses on a compact, reproducible numerical workflow: deterministic parameter defaults, structured output folders, and one or more figures saved for the gallery.

## Outputs

This experiment writes into `out/e072/`:

- `figures/fig_01_race_mod4_diff.png`

## How to run

```bash
make run EXP=e072
```

## Notes

- The gallery preview figure shipped with the documentation uses conservative cutoffs so builds stay fast. If you run the experiment locally, increase the cutoffs to see the asymptotic regime more clearly.
- Prime-race plots depend on the chosen sampling of `x` (linear vs. log grid). The qualitative “who leads” picture can change when you zoom in.

## Published run snapshot

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

```{include} ../reports/e072.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e072.json
:language: json
```

:::

## References

{cite:t}`granvillemartin2006primenumberraces, rubinsteinsarnak1994chebyshevsbias`

## Related experiments

- {doc}`e073` (Prime race mod 3: pi(x;3,2) vs. pi(x;3,1).)
- {doc}`e074` (Prime race mod 8: leaderboard among 1,3,5,7.)
- {doc}`e075` (Prime race statistic: distribution on a log-grid.)
- {doc}`e112` (E112: Prime race: π(x;q,a) − π(x;q,b))
- {doc}`e081` (Prime race sign changes: first crossings table.)