Cyclic Figurate Numbers and Extremal Ellipses in Project Euler's Hardest Problems — E8 Intelligence Research

FINDING: Project Euler hardest unsolved problems involve cyclic figurate numbers and extremal ellipses, with no new universal constants or geometric ratios identified. MATH: Figurate numbers: heptagonal \( P_7(n) = n(5n-3)/2 \); extremal ellipse area/perimeter minimization over fixed points (Euler E563, E691, E692). CONNECTION: No explicit golden ratio, base-60, or crystallographic symmetry found in the provided snippets. Figurate numbers relate to polygonal geometry but not to the specified harmonic ratios. DEPTH: 2 — The findings are algorithmic and historical, not revealing new fundamental mathematical structure or deep geometric constants. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com