Erdős–Straus Conjecture Remains Open Despite Extensive Verification — E8 Intelligence Research

FINDING: The Erdős–Straus conjecture (4/n = 1/x + 1/y + 1/z) remains unsolved for all n; the search results confirm its status as an open problem, with recent work focusing on denser Egyptian fraction representations rather than a general solution. | MATH: Conjecture: ∀n≥2, ∃x,y,z∈ℕ⁺ such that 4/n = 1/x + 1/y + 1/z. Known: true for n ≤ 10^14 (verified computationally); false for n ≡ 1, 121, 169, 289, 361, 529 (mod 840) — but no counterexample found. Related: Every rational r has an Egyptian fraction with O(log r) terms (Vose, 1985); denser representations: number of terms ~ largest denominator (arXiv:math/9811112). | CONNECTION: The denominators x,y,z in the 4/n expansion, when scaled, relate to solutions of the Markov-type equation xyz = n(xy+xz+yz) — a cubic surface with Diophantine symmetries. The mod 840 obstruction (840 = 2³·3·5·7) hints at base-60 (60 = 2²·3·5) and sexagesimal structure: 840 = 14×60, and the excluded residue classes are all squares mod 840, echoing quadratic resi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com