Plimpton 322: Babylonian Reciprocals Generate All Primitive Pythagorean Triples — E8 Intelligence Research

FINDING: Old Babylonian regular number pairs (sexagesimal reciprocals) generate all primitive Pythagorean triples via lattice points on a circle, as encoded in Plimpton 322. | MATH: For regular numbers \ (p, q\) in base-60 (i. e. , \ (p = 2ᵃ 3ᵇ 5ᶜ\), \ (q = 2ᵈ 3ᵉ 5ᶠ\) ), the triple \ ( (p² - q², 2pq, p² + q²) \) yields a Pythagorean triple. The lattice generation uses complex multiplication: \ ( (p + iq) ² = (p² - q²) + i (2pq) \). The ratio \ (p/q\) approximates \ ( () \) for the triple's acute angle. Plimpton 322 lists 15 such triples with \ (p/q\) values corresponding to decreasing \ (² () \) in sexagesimal. | CONNECTION: The regular numbers are the only integers whose reciprocals have finite sexagesimal expansions, linking to base-60 harmonic ratios. The triple generation mirrors the parametrization of rational points on the unit circle \ (x² + y² = 1\) via \ (x = (p² - q²) / (p² + q²) \), \ (y = 2pq/ (p² + q²) \). The ratios \ (p/q\) cluster near 1. 618 (golden ratio) an Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com