Plimpton 322: Ancient Babylonian Table of Reciprocal Pairs Generating Pythagorean Triples — E8 Intelligence Research

FINDING: Plimpton 322 encodes a systematic table of reciprocal pairs (regular sexagesimal numbers) generating Pythagorean triples via a parameterization equivalent to \( (p^2 - q^2, 2pq, p^2 + q^2) \), with columns representing \(\frac{p}{q}\) and \(\frac{p^2+q^2}{2pq}\) in base-60. MATH: - Regular numbers: integers whose only prime factors are 2, 3, 5 — i.e., \(2^a 3^b 5^c\), giving finite sexagesimal reciprocals. - The tablet's columns (per Bruins/Neugebauer reconstruction): Column I: \(\frac{p^2+q^2}{2pq}\) (squared secant or cosecant ratio) Column II: \(\frac{p^2 - q^2}{2pq}\) (squared tangent) Column III: \(\frac{2pq}{p^2+q^2}\) (squared sine) Column IV: row index. - The generating rule: choose \(p > q\), both regular, with \(p/q\) in descending order; then \(a = p^2 - q^2\), \(b = 2pq\), \(c = p^2 + q^2\) form a Pythagorean triple. - Explicit example (row 1): \(p=12, q=5\) → \(a=119, b=120, c=169\) — all in base-60 as 1:59, 2:00, 2:49. - The "missing pa Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com