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., \(p = 2^a 3^b 5^c\), \(q = 2^d 3^e 5^f\)), the triple \((p^2 - q^2, 2pq, p^2 + q^2)\) yields a Pythagorean triple. The lattice generation uses complex multiplication: \((p + iq)^2 = (p^2 - q^2) + i(2pq)\). The ratio \(p/q\) approximates \(\tan(\theta)\) for the triple's acute angle. Plimpton 322 lists 15 such triples with \(p/q\) values corresponding to decreasing \(\tan^2(\theta)\) 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 \(x^2 + y^2 = 1\) via \(x = (p^2 - q^2)/(p^2 + q^2)\), \(y = 2pq/(p^2 + q^2)\). 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