Dense Egyptian Fractions: Improving Erdős–Straus Bounds — E8 Intelligence Research

FINDING: The Erdős–Straus conjecture (4/n = 1/x + 1/y + 1/z) remains unsolved; recent work shows Egyptian fractions can be made "dense" — term count scales with largest denominator, improving prior bounds. | MATH: Conjecture: ∀n>1, ∃x,y,z ∈ ℕ⁺: 4/n = 1/x + 1/y + 1/z. Dense representation result (arXiv:math/9811112v1): For rational r, ∃ Egyptian fraction with k terms and max denominator D such that k = Θ(D) (previously k = o(D) or weaker). Sylvester sequence: s₀=2, sₖ = s₀s₁…sₖ₋₁ + 1; identity: Σᵢ₌₀ⁿ⁻¹ 1/sᵢ + 1/(sₙ−1) = 1. Modular constraint: For 4/n, if n ≡ 1 mod 4, need x ≡ 1 mod 4 etc. — parity and residue restrictions force specific term structures. | CONNECTION: Sylvester sequence grows doubly exponentially; its terms are coprime, and the identity Σ 1/sᵢ → 1 is a canonical Egyptian partition of unity. The ratios sₖ₊₁/sₖ → 1 (not golden), but the *greedy* Egyptian algorithm produces denominators that often relate to continued fractions — and the harmonic series partial sums 1+1/2+…+ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com