{
  "abstract": "FINDING: Euclid-Euler theorem establishes a bijection between even perfect numbers and Mersenne primes, revealing a deep binary-scaling symmetry in the divisor function. | MATH: Even perfect number \\(N = 2^{p-1}(2^p - 1)\\) where \\(2^p - 1\\) is prime (Mersenne prime). Equivalently, \\(\\sigma(N) = 2N\\) (σ = sum of divisors). The binary representation of \\(N\\) is \\(p\\) ones followed by \\(p-1\\) zeros — a self-similar block structure. | CONNECTION: The form \\(2^{p-1}(2^p - 1)\\) exhibits a **geometric ratio** of \\(2^p - 1\\) to \\(2^{p-1}\\), which for large \\(p\\) approaches \\(2\\) — but more strikingly, the **fractional part** of \\(N/(2^p - 1)^2\\) is \\(2^{-(p+1)}\\), linking to dyadic scaling. The **sum of reciprocals of divisors** of a perfect number equals 2 (harmonic mean of divisors is an integer — a lattice-like symmetry). The exponent \\(p\\) itself must be prime, and \\(2^p - 1\\) prime implies \\(p\\) prime — a **prime-chain constraint** analogous to root system rank restrictions in crystallogr\n\nAuthor: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher, United Kingdom"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Andrew Stewart Caldin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7q53cdkwhvza6.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23658255",
  "fulltext_url": null,
  "id": "7q53cdkwhvza6",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23658255",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7q53cdkwhvza6.ris",
  "sources": [
    {
      "first_seen": "2026-11-09T12:21:37Z",
      "native_id": "23658255",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23658255"
    }
  ],
  "subjects": [],
  "title": "Euclid-Euler Theorem: Binary-Scaling Symmetry Links Even Perfect Numbers to Mersenne Primes — E8 Intelligence Research",
  "title_text": "Euclid-Euler Theorem: Binary-Scaling Symmetry Links Even Perfect Numbers to Mersenne Primes — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-09T12:21:37Z",
  "url": "https://research.draconitic.com/paper/7q53cdkwhvza6",
  "venue": "Zenodo",
  "versions": []
}