# E129: Euler lucky constants for $n^2 + n + b$

```{figure} ../_static/experiments/e129_hero.png
:width: 80%
:alt: Preview figure for E129
```

```{figure} ../_static/experiments/e129_hero_2.png
:width: 80%
:alt: Preview figure for E129
```

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

## Highlights

- Compares several classical “lucky” constants $b$ side-by-side.
- Heatmap shows prime/composite patterns across $n$ for each $b$.
- Report includes the first failure $n$ and a factorization witness.

## Goal

Compare how long $n^2+n+b$ stays prime at the start for a small set of classical constants $b$.
We quantify the initial prime streak length and visualize where it breaks.

## Background (quick refresher)

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

## Research question

Among a short list of traditional “Euler lucky numbers” $b$, which yields the longest initial prime streak for
$f(n)=n^2+n+b$ and how do the failures look (first composite and its factors)?

## Method

- Choose a list of constants $b$.
- For each $b$, evaluate $f(n)$ for $n=0..N$.
- Use a sieve to classify values as prime/composite and measure the initial run length.
- Visualize a prime indicator map and a bar chart of run lengths.

## How to run

```bash
make run EXP=e129
```

or:

```bash
uv run python -m mathxlab.experiments.e129
```

## Outputs

This experiment follows the standard output contract:

- `out/e129/figures/` — generated figures (PNG)
- `out/e129/report.md` — short narrative report
- `out/e129/params.json` — run parameters (stable JSON)
- `out/e129/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/e129.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e129.json
:language: json
```

:::

## References

- Lucky numbers and Euler’s polynomial: {cite:t}`WikipediaContributors2025LuckyNumbersOfEuler`, {cite:t}`Weisstein2025EulerPrimeMathWorld`.

## Related experiments

- {doc}`e013` (Euler’s $n^2+n+41$ counterexample witness)
- {doc}`e127` (full (a,b) atlas for $n^2+an+b$)
- {doc}`e128` (modular obstructions that force failures)
- {doc}`e019` (prime density and “random prime” heuristics)