Cyclic Figurate Numbers: A Combinatorial Link Across Polygonal Sequences — E8 Intelligence Research

FINDING: Cyclic set of figurate numbers (triangular, square, pentagonal, hexagonal, heptagonal, octagonal) with polygonal indexing | MATH: Figurate number formulas: \(P_{s,n} = \frac{n}{2}[(s-2)n - (s-4)]\) for \(s=3..8\). Cyclic condition: last two digits of \(P_{s,n}\) equal first two digits of \(P_{s+1,m}\). | CONNECTION: No explicit golden ratio or base-60; figurate numbers relate to polygonal geometry, but cyclic linking is combinatorial, not harmonic. | DEPTH: 4 — elegant combinatorial search, no profound constant or symmetry. FINDING: Prime sum below one million that is itself prime and longest sum of consecutive primes (Problem 50) | MATH: Sum of primes \(S = \sum_{i=k}^{k+L} p_i < 10^6\), \(S\) prime. Algorithm relies on prime density and partial sums. Connection to Riemann Hypothesis: heuristic bound on prime gaps. | CONNECTION: None to geometric harmony. | DEPTH: 5 — touches analytic number theory, but no new constant or symmetry. FINDING: Champernowne constant digit extra Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com