# E050: Primorials and Euclid numbers: $p\#\pm 1$ are usually composite

```{figure} ../_static/experiments/e050_hero.png
:width: 80%
:alt: Preview figure for E050
```

```{figure} ../_static/experiments/e050_hero_2.png
:width: 80%
:alt: Preview figure for E050
```

**Tags:** `number-theory`, `counterexample-search`, `quantitative-exploration`, `visualization`, `primorial`
See: {doc}`../tags`.

## Highlights

- Computes primorials $p_k\#=\prod_{i\le k} p_i$ and Euclid-style numbers $p_k\#\pm 1$.
- Uses a probable-prime test to show “often composite” behavior early.
- Finds small factor witnesses via bounded trial division for many early $k$.

## Goal

Demonstrate that Euclid-style numbers are coprime to small primes but are not reliably prime, and produce explicit factor witnesses.

## Background (quick refresher)

- {doc}`../background/primorials`
- {doc}`../background/prime-numbers`

## Research question

For small $k$, how often are $p_k\#\pm 1$ prime (or probable prime), and how easy is it to find a small factor when they are composite?

## Experiment design

- Compute primorials for $k\le k_{max}$.
- Test $p_k\#\pm 1$ with Miller–Rabin (probable prime).
- If composite under MR, try to find a small trial-division factor as a concrete witness.
- Plot $\log_{10}(p_k\#\pm 1)$ and smallest factor witnesses (log scale).

## Reproducibility

- `params.json` records the run settings.
- `report.md` summarizes the key findings.
- `figures/*.png` contains the plots for the run.

## Interpreting the results

- “Probable prime” is not a proof; it’s a fast filter for this exploratory experiment.
- Factor witnesses come from bounded trial division: not finding a factor is expected for some $k$.
- You should see early composite examples such as $p_6\#+1=30031$ (with a small factor) under default bounds.

## Published run snapshot

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

```{include} ../reports/e050.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e050.json
:language: json
```

:::

## References

See {cite:t}`HardyWright2008AnIntroductionToTheTheoryOfNumbers`, {cite:t}`OEISFoundationInc2025A002110Primorial`, {cite:t}`PrimePagesGlossary2025Primorial`, {cite:t}`WikipediaContributors2025Primorial`.

## Related experiments

- {doc}`e048` (Carmichael numbers: Korselt scan + Fermat counterexamples)
- {doc}`e005` (Odd Perfect Numbers — Constraint Filter Pipeline)
- {doc}`e012` (Fermat pseudoprimes and Carmichael numbers (counterexamples))
- {doc}`e013` (Prime-polynomial counterexamples (Euler's $n^2 + n + 41$))
- {doc}`e014` (Primorial ± 1 counterexamples)