Heptagonal Figurate Numbers, Euler's Ellipse, and Project Euler Optimization — E8 Intelligence Research

FINDING: Cyclic sets of figurate numbers (heptagonal) and algorithmic optimization for Project Euler problems; Euler's extremal ellipse problem from 1770s. MATH: Figurate numbers: heptagonal numbers formula \( P7,n = n(5n-3)/2 \). Problem 50 involves prime sums and Riemann hypothesis connection. Problem 44 uses number theory for pentagonal numbers: \( P5,n = n(3n-1)/2 \). Euler's ellipse: minimal area/perimeter ellipse through fixed points — variational calculus, no explicit constants given. CONNECTION: Figurate numbers relate to polygonal geometry (heptagon, pentagon) but no direct golden ratio or base-60. Euler's ellipse problem touches extremal geometry, not crystallographic symmetry. No 0.382, 0.618, 0.786, 1.618, 2.618, base-60, or root systems evident. DEPTH: 3 — Standard competition problems; algorithmic cleverness but no new universal constants or geometric harmony. Euler's ellipse is historically interesting but not a breakthrough in modern mathemat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com