# E053: Inverse totient multiplicities

```{figure} ../_static/experiments/e053_hero.png
:width: 80%
:alt: Preview figure for E053
```

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

## Highlights

- Compute $\varphi(n)$ for $n\le N$ and count how many preimages each value has.
- Plot the histogram of multiplicities and list the top collisions.

## Goal

Empirically study the many-to-one nature of the totient function.

## Background (quick refresher)

- {doc}`../background/euler-totient-function`
- {doc}`../background/arithmetic-functions`

## Research question

For $n\le N$, how large can the multiplicity of a totient value get, and what do the maximizers look like?

## Method

- Compute $\varphi(n)$ on a range and build a frequency table for values.
- Visualize multiplicities (histogram) and inspect the most frequent values.

## How to run

- `make run EXP=e053`
- `uv run python -m mathxlab.experiments.e053`

## Outputs

This experiment follows the standard output contract:

- `out/e053/figures/` — generated figures (PNG)
- `out/e053/report.md` — short narrative report
- `out/e053/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/e053.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e053.json
:language: json
```

:::

## References

See {cite:t}`apostol1976introanalyticnumbertheory`, {cite:t}`nivenzuckermanmontgomery1991introductiontheorynumbers`.

## Related experiments

- {doc}`e052` (Totient ratio landscape)
- {doc}`e119` (E119: Summatory totient Φ(x) scaling check)
- {doc}`e005` (Odd Perfect Numbers — Constraint Filter Pipeline)
- {doc}`e060` (Jordan totients)
- {doc}`e062` (Carmichael λ(n) vs. φ(n))