Unsolved Math Problems: Hadwiger-Nelson, Josephus, Collatz, Odd Perfect Numbers — E8 Intelligence Research

FINDING: The search results surface four distinct unsolved/classic problems from Numberphile/Veritasium — Hadwiger-Nelson (chromatic number of the plane), Josephus problem, Collatz conjecture, and odd perfect numbers — plus an unrelated omnidirectional video quality paper. | MATH: Hadwiger-Nelson: χ(ℝ²) ∈ {5,6,7} (lower bound 5 via de Grey 2018, upper bound 7 via hexagonal tiling); Josephus: J(n,k) = (J(n−1,k)+k) mod n, closed form J(n,2) = 2l+1 where n = 2^m + l; Collatz: T(n) = n/2 if even, 3n+1 if odd, conjecture ∀n ∃k: T^k(n)=1; odd perfect numbers: σ(N) = 2N, N odd — existence unknown, lower bound N > 10^1500. | CONNECTION: Hadwiger-Nelson upper bound 7 arises from hexagonal tiling — a crystallographic lattice (hexagonal close-packed symmetry, 6-fold rotational symmetry). The lower bound 5 uses Moser spindle (unit-distance graph) and de Grey's 1581-vertex graph — related to lattice/root system geometry (E8-like constraints on unit-distance graphs). Josephus problem has modular ari Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com