E062: Carmichael λ(n) vs. φ(n)

Preview figure for E062
Preview figure for E062

Tags: number-theory, quantitative-exploration, visualization, arithmetic-functions, carmichael-lambda, totient See: Valid 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)

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).

  • n_max: 120000

  • min λ/φ: 0.000772

  • max λ/φ: 1.000000

Figures:

  • fig_01_ratio_lambda_over_phi.png

  • fig_02_log10_lambda_hist.png

params.json (snapshot)
{
  "hist_bins": 60,
  "n_max": 120000
}

References

See Erdős et al. [1991], Montgomery and Vaughan [2006].