Greedy Egyptian Fractions: Linking Modular Inverses, Continued Fractions, and the Erdős–Straus Conjecture — E8 Intelligence Research

FINDING: Egyptian fraction greedy algorithm connects to modular inverses, continued fractions, and lattice reduction — with the Erdős–Straus conjecture as the central unsolved core. | MATH: Erdős–Straus: 4/n = 1/x + 1/y + 1/z (n>1, positive integers). Greedy algorithm: for a/b, choose largest unit fraction ≤ a/b → 1/⌈b/a⌉, then recurse on (a·⌈b/a⌉ − b)/(b·⌈b/a⌉). Modular inverse: solve ax ≡ 1 (mod m) via extended Euclidean algorithm — the greedy step's denominator ⌈b/a⌉ is the ceiling of the continued fraction convergent's denominator. Continued fractions over imaginary quadratic rings (arXiv:1908.00121) generalize this to non-Euclidean fields, retaining exponential convergence — implying the greedy structure is not Euclidean-specific but a deeper lattice phenomenon. | CONNECTION: The greedy step's denominator ⌈b/a⌉ is the ceiling of the continued fraction partial quotient — and the modular inverse's existence condition (gcd(a,m)=1) mirrors the lattice reduction condition for unimodula Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com