Greedy Egyptian Fractions for 4/n: Term Count and Denominator Growth — E8 Intelligence Research

Abstract

FINDING: Erdős–Straus conjecture (4/n = 1/x + 1/y + 1/z for all n≥2) remains unproven; greedy algorithm gives finite but often denominator-bloating representations; denser Egyptian fractions show term-count ~ largest denominator. MATH: Conjecture: ∀n≥2, ∃x,y,z∈ℕ⁺: 4/n = 1/x + 1/y + 1/z. Greedy: for 4/n, first term = ⌈n/4⌉⁻¹, remainder recursion. Known: true for n ≡ 2 mod 4 (trivial via 4/n = 1/n + 1/(n/2) + 1/n? — actually 4/n = 1/(n/2) + 1/n + 1/n fails distinctness; standard: n≡2 mod 4 → 4/n = 1/(n/2) + 1/(n/2+1) + 1/(n(n/2+1))). Denser result (arxiv math/9811112): for any rational r, ∃ Egyptian fraction with number of terms O(log q) and max denominator O(q log q) — but for 4/n specifically, term count 3 is the open question. CONNECTION: No direct geometric ratio (0.382, 0.618, 1.618) appears. However, the structure of unit fractions relates to harmonic series and the Riemann zeta function ζ(1) divergence; the greedy algorithm's denominator growth for 4/n often follows patterns l Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com