Erdős–Straus Conjecture: Unresolved, with Dense Egyptian Fraction Progress — E8 Intelligence Research

FINDING: The Erdős–Straus conjecture (4/n = 1/x + 1/y + 1/z) remains unproven; the search results are videos and a density paper, not a resolution. The only substantive mathematical result is the arXiv paper on dense Egyptian fractions (every rational has representations with term count ~ largest denominator). | MATH: Conjecture: ∀n≥2, ∃x,y,z∈ℕ⁺: 4/n = 1/x + 1/y + 1/z. Known partial results: n ≡ 2 (mod 3) → 4/n = 1/(n) + 1/(n·(n+2)/3) + 1/(n·(n+2)/3·(n+1)/3) type decompositions; n ≡ 1 (mod 4) → 4/n = 1/(n) + 1/(n·(n+1)/4) + 1/(n·(n+1)/4·(n+3)/4). Density result (arXiv:math/9811112v1): for rational r, ∃ Egyptian fraction with k terms and max denominator D such that k = O(log D) (improved to k ~ D^{o(1)}? — the abstract states "same order of magnitude as the largest denominator," i.e., k = Θ(D) in the worst case, which is a strong bound). | CONNECTION: No direct geometric ratio (0.382, 0.618, 1.618) appears. However, the structure of unit fractions relates to harmonic series and the reci Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com