Euler's Legacy: From Heptagonal Numbers to Minimal Ellipses in Project Euler's Core — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
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: P7(n) = (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 = π 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 P2/(4π A) ≥ 1, with equality only for circles. For ellipses through 3 points, the minimal-area ellipse often has eccentricity e = √(1−b2/a2) whose complementary ratio b/a can approach 0.618 (golden ratio conjugate) in degenerate configu
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026eulers,
title = {{Euler's Legacy: From Heptagonal Numbers to Minimal Ellipses in Project Euler's Core — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23658323},
url = {https://zenodo.org/records/23658323},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Euler's Legacy: From Heptagonal Numbers to Minimal Ellipses in Project Euler's Core — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23658323
UR - https://zenodo.org/records/23658323
AB - 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) = (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 = π 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/(4π A) ≥ 1, with equality only for circles. For ellipses through 3 points, the minimal-area ellipse often has eccentricity e = √(1−b^2/a^2) whose complementary ratio b/a can approach 0.618 (golden ratio conjugate) in degenerate configu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -