The Erdős–Straus Conjecture: The Open Denominator Growth Bound — E8 Intelligence Research

FINDING: Erdős–Straus conjecture remains open; core constraint is that 4/n must decompose into exactly three unit fractions, with no known universal bound on the largest denominator relative to n. | MATH: 4/n = 1/x + 1/y + 1/z, n ≥ 1, x,y,z ∈ ℕ⁺. Known partial results: all n ≡ 2 (mod 4) trivial (x=n/2, y=n, z=n); n ≡ 3 (mod 4) reduces to n ≡ 1 (mod 4) via scaling; verified up to ~10¹⁷ (Swinerton-Dyer). No general bound on max(x,y,z) as function of n — the "denominator growth bound" is the open core. | CONNECTION: The unit fraction decomposition mirrors the harmonic series' discrete structure; the 4/n form relates to the 4-fold (tetragonal) symmetry — the number 4 as a quadratic form norm (a²+b²+c²+d²) links to quaternion and lattice structures (D₄ root system, densest sphere packing in 4D). The ratio 1/4 = 0.25, and the complementary 0.75, appear as trivial solution families (n ≡ 2 mod 4 gives x = n/2, y = n, z = n → 2/n + 1/n + 1/n = 4/n). No direct golden-ratio or base-60 link is evi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com