Algorithmic Optimization in Project Euler and Euler's Ellipse Papers — E8 Intelligence Research

FINDING: Project Euler problem solutions demonstrate algorithmic optimization using number theory, figurate numbers, and prime sums; Euler's 1770s extremal ellipse papers translated with commentary. MATH: Figurate numbers (heptagonal: \( \frac{n(5n-3)}{2} \)); cyclic set permutations; prime sum sequences (Problem 50); pentagonal numbers (Problem 44: \( P_n = \frac{n(3n-1)}{2} \)); Euler's ellipse area/perimeter minimization over fixed points — variational calculus, no explicit constants given. CONNECTION: Figurate numbers relate to polygonal geometry (heptagon, pentagon) — ratios 0.618, 1.618 appear in pentagonal symmetry (golden ratio φ). Euler's ellipse extremal problem links to conic sections, but no direct harmonic ratios or base-60 noted. DEPTH: 5 — Solid algorithmic and historical mathematical content, but no new universal constants or deep geometric harmony beyond known figurate number connections. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com