E010: Even perfect numbers from Mersenne primes

Preview figure for E010

Tags: number-theory, quantitative-exploration, visualization See: Valid Tags.

Highlights

  • Generate even perfect numbers via the Euclid–Euler theorem.

  • Connect Mersenne prime exponents \(p\) to perfect numbers \(N = 2^{p-1}(2^p-1)\).

  • Visualize growth and verify the defining property \(\sigma(N) = 2N\) for sampled cases.

Goal

Make the Mersenne \(\leftrightarrow\) perfect-number connection computationally explicit:

If \(M_p = 2^p - 1\) is prime, then:

\[ N_p = 2^{p-1}(2^p-1) \]

is an even perfect number.

Background (quick refresher)

Research question

Across the Mersenne prime exponents discovered within your scan bounds:

  • how fast do the corresponding even perfect numbers grow?

  • can we verify perfectness (\(\sigma(N)=2N\)) efficiently for these cases?

Why this qualifies as a mathematical experiment

  • Finite procedure: find a set of Mersenne primes within a finite bound and generate perfect numbers.

  • Observable(s): size metrics (digits), and validation checks of \(\sigma(N)=2N\).

  • Parameter space: vary the exponent bound and validation depth.

  • Outcome: concrete examples + growth plots that support intuition.

  • Reproducibility: exponents tested and successes recorded in artifacts.

Experiment design

Computation

  • Obtain a list of Mersenne prime exponents \(p\) from a scan (or a fixed list for small \(p\)).

  • For each \(p\), compute \(N_p = 2^{p-1}(2^p-1)\).

  • Verify perfectness for these cases:

    \[ \sigma(N_p) = 2N_p. \]

For modest \(p\), exact computation is feasible; for larger \(p\), report size metrics and skip expensive checks.

Outputs

  • table: \(p\), \(M_p\) size, \(N_p\) size, and validation status

  • plot: \(p\) vs. digits of \(N_p\)

  • optional: prime-factor structure display for small cases

How to run

make run EXP=e010

or:

uv run python -m mathxlab.experiments.e010

Notes / pitfalls

  • Don’t attempt divisor-sum sieves for huge \(N_p\); validation must be bounded and explicit.

  • Clearly separate “constructed from theorem” (conditional on \(M_p\) being prime) from “validated by computation”.

Extensions

  • Cross-link to E002 outputs for perfect numbers and compare growth on the same axes.

  • Explore which parts of the perfectness check can be done symbolically using known factorization.

Published run snapshot

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

Reproduce:

make run EXP=e010

Parameters

  • p_max: 20000

  • max_tests: 800

Mersenne prime exponents found

2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423

Derived even perfect numbers

p

digits(N)

2

1

3

2

5

3

7

4

13

8

17

10

19

12

31

19

61

37

89

54

107

65

127

77

521

314

607

366

1279

770

2203

1327

2281

1373

3217

1937

4253

2561

4423

2663

Notes

  • N = 2^(p-1)·(2^p-1) is perfect iff M_p is prime (Euclid–Euler).

  • For large p, we avoid constructing N explicitly and report its size (digits).

params.json (snapshot)
{
  "max_tests": 800,
  "p_max": 20000
}

References

See References.

[Caldwell, n.d., contributors, 2025, Inc., 2025]