# E128: Quadratic modular obstructions (Euler-type)

```{figure} ../_static/experiments/e128_hero.png
:width: 80%
:alt: Preview figure for E128
```

**Tags:** `number-theory`, `model-checking`, `counterexample-search`, `primes`
See: {doc}`../tags`.

## Highlights

- Makes the “hidden reason” a prime streak must fail: modular obstructions.
- Lists explicit residue classes $n\bmod p$ that force divisibility of $f(n)$.
- Includes the classic witness $n=41k$ for Euler’s $n^2+n+41$.

## Goal

Show how congruences explain why a “prime-generating polynomial” cannot stay prime forever.
For each small prime modulus $p$, we compute the roots of $f(n)\equiv 0\pmod p$.

## Background (quick refresher)

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

## Research question

For Euler’s $f(n)=n^2+n+41$, which small primes $p$ divide some values $f(n)$,
and which residue classes $n\bmod p$ generate those guaranteed composites?

## Method

- Fix $f(n)=n^2+an+b$ with $(a,b)=(1,41)$.
- For each prime $p\le p_{\max}$, brute-force all residues $n\in\{0,\dots,p-1\}$ and record solutions to $f(n)\equiv 0\pmod p$.
- Plot the number of roots per prime and write a report table of residues.

## How to run

```bash
make run EXP=e128
```

or:

```bash
uv run python -m mathxlab.experiments.e128
```

## Outputs

This experiment follows the standard output contract:

- `out/e128/figures/` — generated figures (PNG)
- `out/e128/report.md` — short narrative report
- `out/e128/params.json` — run parameters (stable JSON)
- `out/e128/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/e128.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e128.json
:language: json
```

:::

## References

- Euler’s quadratic and its folklore: {cite:t}`WikipediaContributors2025LuckyNumbersOfEuler`, {cite:t}`Weisstein2025EulerPrimeMathWorld`.
- Quadratic basics: {cite:t}`WikipediaContributors2026QuadraticEquation`.

## Related experiments

- {doc}`e013` (first explicit counterexample $f(40)=41^2$)
- {doc}`e031` (admissibility / modular obstructions in prime constellations)
- {doc}`e035` (primes in arithmetic progressions)
- {doc}`e038` (Bertrand’s postulate: primes in intervals)