Plimpton 322: A Secant-Squared Table of Regular Reciprocal Pairs for Pythagorean Triples — E8 Intelligence Research

FINDING: Plimpton 322 contains 15 pairs of regular sexagesimal numbers (reciprocal pairs) used to generate Pythagorean triples via the OB scribal algorithm: take \( p, q \) regular (divisors of 60^k), then triple = \( (p^2 - q^2, 2pq, p^2 + q^2) \). The tablet lists \( (p/q)^2 \) values in descending order, effectively a secant-squared table. MATH: - Regular numbers: \( 2^a 3^b 5^c \) (divisors of 60^k). - Reciprocal pair generation: \( p, q \) such that \( p · q = 60^k \) (or \( p/q \) rational with terminating sexagesimal expansion). - Triple: \( a = p^2 - q^2,\; b = 2pq,\; c = p^2 + q^2 \). - Column I gives \( (c/a)^2 = (p/q + q/p)^2 / 4 \), i.e., \( ^2 θ \) for angle \( θ \) opposite side \( a \). - Ratios on tablet: e.g., line 1: \( p/q ≈ 1.9834 \) → \( (p/q)^2 ≈ 3.934 \), corresponding to angle ~45°. CONNECTION: - **Golden ratio link**: The regular number pairs \( p, q \) produce ratios \( p/q \) that cluster near 1.618 (golden ratio) f Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com