# E127: Quadratic prime-run atlas ($n^2 + a n + b$)

```{figure} ../_static/experiments/e127_hero.png
:width: 80%
:alt: Preview figure for E127
```

```{figure} ../_static/experiments/e127_hero_2.png
:width: 80%
:alt: Preview figure for E127
```

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

## Highlights

- Heatmap of “how long primes last” for $f(n)=n^2+an+b$ across a small (a,b) grid.
- Euler’s $(a,b)=(1,41)$ appears as a visible peak, but it is not unique.
- Turns a folklore fact into a reproducible, parameterized sweep with saved artifacts.

## Goal

Explore how sensitive **initial prime streaks** are to the coefficients of a quadratic polynomial.
We measure, for each (a,b), the largest $L$ such that $f(0),f(1),\dots,f(L-1)$ are all prime.

## Background (quick refresher)

- {doc}`../background/quadratic-polynomials`
- {doc}`../background/eulers-prime-generating-polynomial`
- {doc}`../background/prime-numbers`

## Research question

Within a bounded coefficient grid, which quadratic polynomials produce the **longest initial prime runs**,
and where does Euler’s polynomial sit in that landscape?

## Method

- Sweep integers $a\in[a_{\min},a_{\max}]$ and $b\in[b_{\min},b_{\max}]$.
- For each pair, evaluate $f(n)$ for $n=0..N$ and record the initial prime-run length.
- Visualize run length as a heatmap and summarize the top candidates in the report.

### Plot axes and coordinate conventions

The heatmap uses a 2D grid over the integer coefficients $(a,b)$:

- **x-axis:** the coefficient $a$ (horizontal direction)
- **y-axis:** the coefficient $b$ (vertical direction)

If the sweep uses unit steps (the default), then each heatmap cell corresponds directly to an integer pair:
$$(a,b) \in \{a_{\min},\dots,a_{\max}\} \times \{b_{\min},\dots,b_{\max}\}.$$

More generally, if the heatmap array is indexed by $(i,j)$ with
$i=0,\dots,n_a-1$ (columns) and $j=0,\dots,n_b-1$ (rows), then the coefficients shown at that cell are:

- $$x = a = a_{\min} + i\,\Delta a$$
- $$y = b = b_{\min} + j\,\Delta b$$

For Matplotlib `imshow`, this is typically implemented via an `extent` so that tick labels match the coefficient values
(e.g. `extent=(a_min-0.5, a_max+0.5, b_min-0.5, b_max+0.5)` for unit steps). If you flip the image origin (`origin="upper"`),
the y-axis is inverted visually; in that case, interpret the y-axis tick labels accordingly.

## How to run

```bash
make run EXP=e127
```

or:

```bash
uv run python -m mathxlab.experiments.e127
```

## Outputs

This experiment follows the standard output contract:

- `out/e127/figures/` — generated figures (PNG)
- `out/e127/report.md` — short narrative report
- `out/e127/params.json` — run parameters (stable JSON)
- `out/e127/logs/` — run logs (created by the runner/Makefile)

## Published run snapshot

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

```{include} ../reports/e127.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e127.json
:language: json
```

:::

## References

- Euler’s quadratic and “lucky numbers”: {cite:t}`WikipediaContributors2025LuckyNumbersOfEuler`, {cite:t}`Weisstein2025EulerPrimeMathWorld`.
- Quadratic basics (discriminant, roots): {cite:t}`WikipediaContributors2026QuadraticEquation`.

## Related experiments

- {doc}`e013` (Euler’s polynomial: first counterexample witness)
- {doc}`e024` (Ulam spiral: prime-rich diagonals explained by quadratic forms)
- {doc}`e031` (modular obstructions and admissibility)
- {doc}`e125` (Sacks spiral prime structure)
- {doc}`e126` (hex-grid spiral prime structure)