FINDING: Old Babylonian sexagesimal reciprocal pairs algorithm for generating Pythagorean triples, as encoded on Plimpton 322. MATH: For reciprocal pair \ ( (p, q) \) in base-60 with \ (p > q\) and \ (p q = 1\) (or scaled to 1), triple \ ( (a, b, c) \) is generated by: \ (a = p - q\), \ (b = 2\) (or \ (p + q\) depending on convention), \ (c = p + q\) (or \ (p - q\) respectively), then scaled by a common factor. Equivalently, using modern terms: let \ (x = p/q\) with \ (x > 1\), then triple sides are \ ( (x - 1/x, 2, x + 1/x) \) normalized. The tablet lists 15 triples with decreasing \ (b²/a²\) ratios, corresponding to decreasing angles from ~45° to ~30°. CONNECTION: The reciprocal pairs inherently involve ratios of the form \ (p/q\) that are rational approximations of \ (2\) and other square roots. The sequence of \ (b²/a²\) values on Plimpton 322 approximates a linear progression in sexagesimal, hinting at a trigonometric table of secant or tangent values. The ratios 0. 618, 1. 618 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com
Andrew Stewart Caldin (Thu,) studied this question.