Collatz Conjecture and Odd Perfect Numbers: Two Unsolved Number Theory Problems — E8 Intelligence Research

FINDING: Collatz Conjecture — iterative function on integers, no proof of termination for all inputs. | MATH: \( f(n) = \begin{cases} n/2 & \text{if } n \text{ even} \\ 3n+1 & \text{if } n \text{ odd} \end{cases} \); no closed-form solution or known invariant. | CONNECTION: None — no geometric ratio, symmetry, or constant emerges from the iteration. | DEPTH: 2 (famous but mathematically barren in terms of harmonic or structural insight). FINDING: Odd Perfect Numbers — no example known, no proof of non-existence. | MATH: \(\sigma(n) = 2n\) for perfect \(n\); odd case requires \(\sigma(n)\) odd, implying \(n\) must be a square times a prime power; lower bound > \(10^{1500}\). | CONNECTION: None — no ratio or symmetry links to known geometric constants. | DEPTH: 1 (purely number-theoretic, no geometric or physical resonance). FINDING: 100 Prisoners Riddle — optimal strategy uses cycle decomposition of permutations. | MATH: Probability of success = \(1 - \sum_{k=\lfloor n/2 \rfloor+1}^n Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com