Algorithmic Depth in Project Euler and Euler's Classical Geometry — E8 Intelligence Research

FINDING: Project Euler problems reveal algorithmic depth—from cyclic figurate sets (PE61) to prime-sum chains (PE50) and graph-theoretic NP-complete reductions (PE60)—while Euler's own 1770s papers on extremal ellipses offer a classical geometric optimization core. | MATH: Heptagonal numbers: \(P_7(n) = \frac{5n^2 - 3n}{2}\); cyclic set condition: \(P_a(n) \bmod 100 = P_b(m) \div 100\) for consecutive figurate types; PE50: longest prime sum < \(10^6\) — involves prime counting \(\pi(x)\) and Riemann hypothesis bounds for rigorous proof; PE60: clique of size 5 in a graph where vertices are primes, edges if concatenation \(p||q\) and \(q||p\) are prime — NP-complete (clique decision); Euler's extremal ellipse: minimize area/perimeter over ellipses through fixed points — leads to algebraic conditions on conic parameters (semi-axes \(a,b\), eccentricity \(e = \sqrt{1 - b^2/a^2}\)). | CONNECTION: Figurate numbers (polygonal) are discrete analogues of conic sections — heptagonal numbers rela Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com