Greedy Egyptian Fractions: Canonical Decomposition and Modular Arithmetic Ties — E8 Intelligence Research

FINDING: The greedy algorithm for Egyptian fractions (Sylvester's sequence) provides a canonical decomposition of rationals into distinct unit fractions, with deep ties to modular arithmetic and lattice reduction; the Fermilab lattice QCD result is a separate, unrelated application of lattice methods to particle masses. MATH: - Greedy algorithm: For \( \frac{a}{b} \) (with \( a < b \)), choose largest unit fraction \( \frac{1}{\lceil b/a \rceil} \), subtract, repeat. Terminates for all rationals. - Sylvester's sequence: \( s_0 = 2 \), \( s_{n+1} = s_1 s_2 \cdots s_n + 1 \). The greedy expansion of \( \frac{1}{s_0} + \frac{1}{s_1} + \cdots \) yields \( \frac{1}{2} + \frac{1}{3} + \frac{1}{7} + \frac{1}{43} + \cdots \). - Key identity: \( \frac{1}{s_n} = \frac{1}{s_{n+1}-1} - \frac{1}{s_{n+1}} \), giving telescoping sums. - Modular structure: The denominators \( s_n \) satisfy \( s_{n+1} \equiv 1 \pmod{s_n} \), and \( s_{n+1} \equiv 1 \pmod{s_i} \) for all \( i \le n \). This is Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com