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

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 \(p \cdot q = 1\) (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 \(b^2/a^2\) 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 \(\sqrt{2}\) and other square roots. The sequence of \(b^2/a^2\) 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