The Transparent Egyptian Fraction Theorem (2025 Edition)

This work presents a fully transparent, minimal, and rigorously elementary proof of the classical Egyptian Fraction Theorem: every rational number (0 < a/b < 1) can be expressed as a finite sum of distinct unit fractions. Although the theorem is nearly 800 years old, dating back to Fibonacci’s Liber Abaci (1202), a proof written in a genuinely minimal “canonical form” has been surprisingly absent from the literature. This paper rediscovered and refined the classical greedy algorithm into its most concise modern structure, showing that the numerator strictly decreases at every non-terminating step. This single observation forces the algorithm to terminate by infinite descent on the positive integers. As a result, the entire theorem can be proved using only two lines of inequalities and one structural invariant. The goal of this paper is not to introduce a new theorem, but to present the clearest possible modern formulation of an important and historically rich result in elementary number theory. This version is intended to serve as a clean reference point for students, educators, and researchers seeking the shortest rigorous proof of the Egyptian fraction decomposition. This publication also marks the beginning of my formal research output in number theory and foundational mathematics. Author: June Woo KooDate: 2025Keywords: Egyptian Fraction, Greedy Algorithm, Rational Numbers, Infinite Descent, Number Theory, Classical Proofs