Erdős–Straus Conjecture: Open Status and Modular Reduction to 840 Residues — E8 Intelligence Research

FINDING: The Erdős–Straus conjecture (4/n = 1/x + 1/y + 1/z) remains unresolved; the search results confirm its status as an open problem, with the only rigorous progress being modular constraints (n ≡ 1, 121, 169, 289, 361, 529 mod 840 are the only residues requiring proof). The geometric-series video is unrelated to the conjecture. | MATH: Conjecture: ∀n≥2, ∃x,y,z∈ℕ⁺ s.t. 4/n = 1/x + 1/y + 1/z. Known: n ≡ 0 mod 4 trivial; n ≡ 2 mod 4 → 4/n = 1/n + 1/(n/2) + 1/n; n ≡ 3 mod 4 → 4/n = 1/n + 1/((n+1)/2) + 1/(n(n+1)/2). Remaining hard cases: n ≡ 1 mod 4 with n ≡ 1, 121, 169, 289, 361, 529 mod 840. No constants or ratios emerge from the conjecture itself. | CONNECTION: The modulus 840 = 2³·3·5·7 — this is the order of the largest exceptional Lie group's Weyl group? No — 840 is |W(E₇)|/2 = 2903040/3456? Actually 840 = 2³·3·5·7, and it is the smallest number with 32 divisors. It relates to the crystallographic root system E₇'s Coxeter number (18) and the 240 roots of E₈? No direct link. Howe Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com