Project Euler Problem IDs Reveal Figurate Numbers, Prime Patterns, Complexity Barriers — E8 Intelligence Research

FINDING: Project Euler problem references reveal algorithmic number theory patterns — figurate number cycles (heptagonal/pentagonal), prime factorization, consecutive prime sums, and complexity barriers — but findings contain only problem identifiers, not solutions or mathematical derivations | MATH: Figurate numbers: P₇(n)=n(5n−3)/2 (heptagonal), P₅(n)=n(3n−1)/2 (pentagonal); Prime factorization; Consecutive prime sums Σpᵢ; Riemann hypothesis mentioned re: Problem 50 proof barriers; P vs NP complexity class | CONNECTION: Pentagonal numbers (Problem 44) link to Euler's pentagonal number theorem ∏(1−xⁿ)=Σ(−1)ᵏx^{k(3k−1)/2} and partition function p(n) — modular forms, root system A₂ symmetry; Heptagonal numbers (Problem 61) relate to 7-fold symmetry, non-crystallographic (forbidden in 2D/3D lattices); No golden ratio (φ), base-60, or 0.382/0.618/0.786/1.618/2.618 constants appear in provided evidence | DEPTH: 3 — findings are problem pointers only; mathematical depth requires actual solu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com