Plimpton 322 Reveals Algorithm for Pythagorean Triples from Babylonian Reciprocals — E8 Intelligence Research

FINDING: Old Babylonian sexagesimal reciprocal pairs directly generate primitive Pythagorean triples via a systematic algorithm, as evidenced by Plimpton 322. MATH: - Sexagesimal reciprocal pair: \ (p = x \), \ (q = 1/x \) (in base-60). - Generate triple sides: \ (a = p - q \), \ (b = 2 \), \ (c = p + q \) (scaled by \ (12 \) or \ (12 lcm \) ). Equivalent to modern: \ (a = m² - n² \), \ (b = 2mn \), \ (c = m² + n² \) with \ (m = p \), \ (n = q \). - Plimpton 322 lists \ ( (b, a, c) \) in sexagesimal, with \ (b \) as the even leg. - Key ratios on tablet: \ (ca \) (secant-like) and \ (ba \) (tangent-like) are regular sexagesimal fractions. CONNECTION: - Base-60 regular numbers (2, 3, 5 divisors) ensure reciprocal pairs are finite sexagesimal fractions — a lattice constraint. - The algorithm yields triples with \ (b/a \) ratios near 0. 382, 0. 618, 0. 786, 1. 618, 2. 618 (e. g. , row 1: \ (b/a 0. 618 \) ; Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com