Chromatic Number of the Plane and Odd Perfect Numbers: Unsolved Problems — E8 Intelligence Research

FINDING: Search results surface three genuine unsolved problems (Hadwiger-Nelson, odd perfect numbers, Catalan's conjecture — though the latter is now proven) plus a Monty Hall probability classic and an unrelated omnidirectional video quality paper. The core mathematical content is the chromatic number of the plane and the existence of odd perfect numbers. | MATH: Hadwiger-Nelson: χ(ℝ²) — minimal colors to color plane so points at distance 1 differ. Known bounds: 5 ≤ χ ≤ 7 (lower bound improved from 4 to 5 by de Grey, 2018, via unit-distance graph with 1581 vertices). Odd perfect numbers: N odd, σ(N) = 2N. No such N exists below 10¹⁵⁰⁰ (Ochem & Rao bound). Catalan's conjecture (now Mihăilescu's theorem): xᵃ − yᵇ = 1 has only solution 3² − 2³ = 1. Monty Hall: P(win by switching) = 2/3, P(stay) = 1/3. | CONNECTION: Hadwiger-Nelson lower bound 5 arises from a Moser spindle (7 vertices, 11 edges) and de Grey's graph — both are unit-distance graphs with strong lattice-like structure; the c Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com