Babylonian Sexagesimal Algorithm for Pythagorean Triples on Plimpton 322 — E8 Intelligence Research

FINDING: Babylonian algorithm for generating Pythagorean triples using sexagesimal (base-60) arithmetic, equivalent to the modern formula \( p^2 - q^2, 2pq, p^2 + q^2 \), as evidenced by Plimpton 322 and related cuneiform tablets. | MATH: For integers \( p > q > 0 \), triple: \( a = p^2 - q^2 \), \( b = 2pq \), \( c = p^2 + q^2 \). Sexagesimal representation uses base-60 place value. Example from Plimpton 322: triple (119, 120, 169) corresponds to \( p=12, q=5 \) in sexagesimal (12 = 0;12, 5 = 0;05). | CONNECTION: The ratio \( b/a = 2pq/(p^2 - q^2) \) yields values near 1 (e.g., 120/119 ≈ 1.0084) and can be expressed as tangents of angles. The generating numbers \( p, q \) often relate to regular sexagesimal numbers (divisors of 60), linking to base-60 harmonic ratios. The triple (3,4,5) corresponds to \( p=2, q=1 \), giving ratio 4/3 ≈ 1.333, not directly a golden ratio, but the method itself embodies a symmetry in integer lattices (the integer grid of \( Z^2 \)). | DEPTH: 7 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com