{
  "abstract": "FINDING: Project Euler problems cluster around discrete number theory, graph theory, and extremal geometry — with the deepest link being Euler's own 1770s work on minimal-area/perimeter ellipses through fixed points. | MATH: Heptagonal numbers: \\(P_7(n) = \\frac{n(5n-3)}{2}\\); Problem 60 uses clique-finding (NP-complete) on prime-sum graphs; Problem 50 involves prime sums with Riemann-zeta-adjacent heuristics; Euler's ellipse problem: minimize area \\(A = \\pi ab\\) or perimeter \\(P = 4aE(e)\\) (complete elliptic integral of 2nd kind) over ellipses through \\(k\\) fixed points — a variational problem with no closed-form general solution. | CONNECTION: The ellipse extremal problem connects to the golden ratio via the *isoperimetric* ratio \\(P^2/(4\\pi A) \\ge 1\\), with equality only for circles. For ellipses through 3 points, the minimal-area ellipse often has eccentricity \\(e = \\sqrt{1-b^2/a^2}\\) whose complementary ratio \\(b/a\\) can approach 0.618 (golden ratio conjugate) in degenerate configu\n\nAuthor: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com",
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  "affiliations": [
    "Independent Researcher, United Kingdom"
  ],
  "article_type": null,
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  "author_total": 1,
  "authors": [
    {
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      ],
      "name": "Andrew Stewart Caldin"
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  "bibtex": "https://research.draconitic.com/paper/7p17x8d0vav3f.bib",
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  "doi": "10.5281/zenodo.23658323",
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  "id": "7p17x8d0vav3f",
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  "published": "2026-10-26",
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  "ris": "https://research.draconitic.com/paper/7p17x8d0vav3f.ris",
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      "first_seen": "2026-11-09T12:21:37Z",
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  "title": "Euler's Legacy: From Heptagonal Numbers to Minimal Ellipses in Project Euler's Core — E8 Intelligence Research",
  "title_text": "Euler's Legacy: From Heptagonal Numbers to Minimal Ellipses in Project Euler's Core — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-09T12:21:37Z",
  "url": "https://research.draconitic.com/paper/7p17x8d0vav3f",
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