From Graph Cliques to Minimal Ellipses: Euler's Dual Mathematical Legacy — E8 Intelligence Research

Abstract

FINDING: Project Euler's hardest problems (PE#60, cyclic figurate sets) are graph-theoretic NP-complete searches; Euler's 1770s extremal ellipse papers (E563, E691, E692) solve minimal-area/perimeter ellipses through fixed points. | MATH: PE#60: prime pairs forming clique in graph — NP-complete (clique decision); figurate numbers: P(s,n) = ((s−2)n² − (s−4)n)/2, cyclic concatenation constraint; Euler's ellipses: minimal area/perimeter ellipse through N points — variational problem, Euler–Lagrange equations, algebraic degree of solution; no closed form for N>5, elliptic integrals appear. | CONNECTION: Ellipse minimality links to conic sections — degenerate cases yield circles (ratio 1:1); extremal ellipses through symmetric point sets (e.g., regular polygons) recover crystallographic point-group symmetries (D_n, dihedral); figurate numbers P(s,n) for s=3,4,5,6,7,8 map to triangular, square, pentagonal, hexagonal, heptagonal, octagonal — these are root-system polytope counts (A_n, B_n, H_ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com