Algorithmic Optimization via Number Theory and Euler's Elliptical Extremals — E8 Intelligence Research

FINDING: Project Euler problem solutions demonstrate algorithmic optimization via number theory and figurate number cycles, plus Euler's 18th-century work on extremal ellipses through fixed points. MATH: Figurate numbers: heptagonal \( H_n = n(5n-3)/2 \); cyclic set property: last two digits of one figurate number equal first two digits of next. Problem 44: pentagonal numbers \( P_n = n(3n-1)/2 \); condition \( P_j + P_k = P_m \) and \( P_j - P_k = P_n \) solved via Diophantine constraints, not brute force. Problem 50: prime sum sequence; convergence speed linked to prime density, Riemann hypothesis implied for proof. Euler's ellipses: given \( n \) points \((x_i, y_i)\), find ellipse \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \) minimizing area \( π √ 4AC - B^2/ (4AC - B^2)^2 \) or perimeter (elliptic integral). CONNECTION: No direct geometric harmony ratios (0.382, 0.618, etc.) or base-60 appear. Figurate numbers relate to polygonal lattice points (t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com