Euler's Ellipse Problem and Algorithmic Number Theory in Project Euler — E8 Intelligence Research

Abstract

FINDING: Project Euler problems reveal algorithmic number theory patterns, but the deepest mathematical link is Euler's 1770s work on extremal ellipses through fixed point sets. | MATH: Cyclic figurate sets (polygonal numbers: P(s,n)=((s-2)n²-(s-4)n)/2); prime-sum chains (Problem 50, Riemann hypothesis connection); Champernowne constant (Problem 40, C10=0.123456789101112...); pentagonal number theorem (Problem 44, P(n)=n(3n-1)/2, generalized pentagonal numbers ±n(3n±1)/2); Euler's ellipse problem: minimize area/perimeter of ellipse through N fixed points — variational problem leading to algebraic conditions on ellipse parameters (semi-axes a,b, rotation θ). | CONNECTION: The pentagonal numbers in Problem 44 connect to the Dedekind eta function and modular forms — whose Fourier coefficients encode the Leech lattice (24-dimensional, related to the golden ratio via its automorphism group Conway group Co0, order 8,315,553,613,086,720,000 = 2²²·3⁹·5⁴·7²·11·13·23). The ellipse extremal probl Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com