Greedy Egyptian Fractions: Super-Exponential Denominators and Harmonic Ties — E8 Intelligence Research

FINDING: The greedy algorithm for Egyptian fractions (Fibonacci–Sylvester) generates unique unit-fraction decompositions whose denominators grow super-exponentially, with deep ties to harmonic series divergence and ancient base-60/unit-fraction arithmetic. | MATH: For rational \(a/b \in (0,1)\), greedy step: \( \frac{a}{b} \to \frac{1}{\lceil b/a \rceil} + \frac{a'}{b'} \), where \(a' = a\lceil b/a \rceil - b\), \(b' = b\lceil b/a \rceil\). Denominators satisfy \(q_{n+1} \ge q_n(q_n - 1) + 1\) (Sylvester's sequence growth). Harmonic series \(\sum_{n=1}^\infty 1/n\) diverges (Oresme's proof: \(1 + 1/2 + (1/3+1/4) + \dots > 1 + 1/2 + 1/2 + \dots\)), yet Egyptian fractions always converge to rationals — the greedy algorithm exploits the *slow* divergence of harmonic series to pack unit fractions into any rational. | CONNECTION: The greedy algorithm's denominator growth \(q_{n+1} \approx q_n^2\) mirrors the *golden ratio conjugate* \(0.618\) in the sense that Sylvester's sequence \(s_n = s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com