Unsolved Mathematical Conjectures: Collatz, Riemann, Poincaré, and Erdős–Straus — E8 Intelligence Research

Abstract

FINDING: The corpus centers on the Collatz Conjecture, the Riemann Hypothesis, the Poincaré Conjecture, and the Erdős–Straus Conjecture, with the Collatz problem as the most frequently cited "simple but unsolved" case. | MATH: Collatz map: \( T(n) = n/2 \) if \( n \) even, \( T(n) = 3n+1 \) if \( n \) odd; conjecture: \( \forall n \in \mathbb{N}, \exists k: T^k(n)=1 \). Riemann Hypothesis: \( \zeta(s)=0 \implies \Re(s)=1/2 \). Poincaré: every simply connected closed 3-manifold is homeomorphic to \( S^3 \). Erdős–Straus: \( \frac{4}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z} \) has positive integer solutions for all \( n \geq 2 \). | CONNECTION: Collatz dynamics exhibit a logarithmic growth ratio near \( \log_2(3) \approx 1.585 \) (not a golden ratio, but a base-2/base-3 tension). The Riemann zeta zeros align with a 1/2 critical line — a symmetry axis analogous to a reflection plane in crystallographic root systems (e.g., \( A_1 \) reflection). The Poincaré sphere \( S^3 \) is the unit qu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com