Number-Theoretic Shortcuts in Project Euler Yield Speedups, No New Constants — E8 Intelligence Research

FINDING: Project Euler problem solutions often exploit number-theoretic shortcuts (e.g., pentagonal number theorem, prime sum sieves) to achieve dramatic speedups, but no new universal constants or geometric ratios emerge. | MATH: Pentagonal numbers: P_n = n(3n-1)/2; Heptagonal numbers: H_n = n(5n-3)/2; Prime sum problem 50 involves Riemann zeta function ζ(s) for convergence bounds. | CONNECTION: None. Figurate numbers relate to polygonal geometry but no golden ratio, base-60, or crystallographic symmetry appears. | DEPTH: 3 — Competence in algorithmic number theory, but no fundamental mathematical discovery. FINDING: Euler's 1770s papers on extremal ellipses (E563, E691, E692) find minimal-area and minimal-perimeter ellipses through fixed point sets, a classical optimization problem. | MATH: For points (x_i, y_i), ellipse equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 with constraint B² - 4AC < 0. Minimize area = π / sqrt(AC - B²/4) or perimeter via elliptic integrals. | CONNECTION: Elli Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com