# E062: Carmichael λ(n) vs. φ(n)

```{figure} ../_static/experiments/e062_hero.png
:width: 80%
:alt: Preview figure for E062
```

```{figure} ../_static/experiments/e062_hero_2.png
:width: 80%
:alt: Preview figure for E062
```

**Tags:** `number-theory`, `quantitative-exploration`, `visualization`, `arithmetic-functions`, `carmichael-lambda`, `totient`
See: {doc}`../tags`.

## Highlights

- Plot or histogram the ratio $\varphi(n)/\lambda(n)$.
- Show how 2-adic structure affects λ(n) (notably for powers of 2).

## Goal

See how the exponent of the multiplicative group mod n differs from its size.

## Background (quick refresher)

- {doc}`../background/carmichael-lambda-function`
- {doc}`../background/euler-totient-function`

## Research question

How large can $\varphi(n)/\lambda(n)$ get for $n\le N$, and what kinds of n cause large gaps?

## Method

- Compute φ(n) and λ(n) from prime-power formulas and lcm composition.
- Visualize ratios (histograms, quantiles) and list top outliers.

## How to run

- `make run EXP=e062`
- `uv run python -m mathxlab.experiments.e062`

## Outputs

This experiment follows the standard output contract:

- `out/e062/figures/` — generated figures (PNG)
- `out/e062/report.md` — short narrative report
- `out/e062/manifest.json` — snapshot metadata for the gallery

## Published run snapshot

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

```{include} ../reports/e062.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e062.json
:language: json
```

:::

## References

See {cite:t}`erdospomeranceschmutz1991carmichaelslambdafunction`, {cite:t}`montgomeryvaughan2006multiplicativenumbertheoryi`.

## Related experiments

- {doc}`e100` (E100: Carmichael λ(n) vs. Euler φ(n))
- {doc}`e052` (Totient ratio landscape)
- {doc}`e053` (Inverse totient multiplicities)
- {doc}`e060` (Jordan totients)
- {doc}`e099` (E099: Jordan totients J_k: identities and ratios)