# E051: Semiprimes: balanced vs. unbalanced factorization timing

```{figure} ../_static/experiments/e051_hero.png
:width: 80%
:alt: Preview figure for E051
```

```{figure} ../_static/experiments/e051_hero_2.png
:width: 80%
:alt: Preview figure for E051
```

**Tags:** `number-theory`, `quantitative-exploration`, `visualization`, `factorization`, `semiprime`
See: {doc}`../tags`.

## Highlights

- Generates small semiprimes $n=pq$ in balanced and unbalanced regimes.
- Factors them using trial division + Pollard’s rho and records timings.
- Visualizes time distributions to make “balanced is harder” tangible.

## Goal

Compare how factorization difficulty changes when one prime factor is much smaller than the other.

## Background (quick refresher)

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

## Research question

For semiprimes of similar overall size, how does Pollard-rho timing differ between balanced $(p\approx q)$ and unbalanced $(p\ll q)$ cases?

## Experiment design

- Generate balanced semiprimes with roughly equal factor bit lengths.
- Generate unbalanced semiprimes with a fixed small factor size.
- Factor each using trial division pre-pass + Pollard’s rho, measuring wall-clock time.
- Plot boxplots and a size-vs-time scatter.

## Reproducibility

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

## Interpreting the results

- Timings are noisy: random instance structure matters a lot at this size range.
- The trend should still show unbalanced instances often factoring faster (small factor is easier to find).
- This is an educational-scale experiment, not a cryptographic benchmark.

## Published run snapshot

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

```{include} ../reports/e051.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e051.json
:language: json
```

:::

## References

See {cite:t}`RivestShamirAdleman1978RSA`, {cite:t}`OEISFoundationInc2025A001358Semiprimes`.

## Related experiments

- {doc}`e004` (Computing $\sigma(n)$ at Scale — Sieve vs. Factorization)
- {doc}`e002` (Even Perfect Numbers — Generator and Growth)
- {doc}`e003` (Abundancy Index Landscape)
- {doc}`e007` (Mersenne growth (bits and digits))
- {doc}`e008` (Lucas–Lehmer scan (prime exponents))