Euler's Legacy: From Heptagonal Numbers to Minimal Ellipses in Project Euler's Core — E8 Intelligence Research

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 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com