Babylonian Reciprocal-Pair Algorithm for Pythagorean Triples in Plimpton 322 — E8 Intelligence Research

FINDING: Babylonian sexagesimal algorithm for generating Pythagorean triples, as encoded in Plimpton 322, uses a reciprocal-pair method with geometric mean approximations. MATH: - Generating formula: For reciprocal pairs \( p \) and \( q \) (with \( p > q \), both regular sexagesimal numbers), triple is: \( a = p^2 - q^2 \), \( b = 2pq \), \( c = p^2 + q^2 \). - On Plimpton 322, the listed values correspond to \( (c/a)^2 \) or \( (b/a) \) ratios, not the triples themselves. - Key constants: - \( √2 ≈ 1;24,51,10 \) (sexagesimal) = \( 1 + 24/60 + 51/3600 + 10/216000 ≈ 1.41421296 \). - Geometric mean approximation: \( √a · b ≈ (a+b)/2 \) for near-equal numbers, refined by \( √a · b ≈ (a+b)/2 - (a-b)^2/(8(a+b)) \). - Base-60 regular numbers used: only those whose reciprocals terminate in sexagesimal (e.g., 2,3,5 divisors). CONNECTION: - Ratios in Plimpton 322 approximate \( √2 \), \( √3 \), and \( √{5 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com