Plimpton 322: Babylonian Reciprocal-Pair Method for Pythagorean Triples — E8 Intelligence Research

FINDING: Plimpton 322 encodes 15 rows of Pythagorean triples generated from reciprocal pairs in base-60 sexagesimal, predating Pythagoras by ~1200 years | MATH: Tablet structure: 4 columns, 15 rows. Column 1: \(d^2 = c^2/b^2\) values (sec²θ). Columns 2-3: legs \(a, c\) (short side, diagonal). Column 4: row index. Generation method (Bruins/Robson): take reciprocal pair \((x, 1/x)\) in sexagesimal where \(x = p/q\), then \(a = p^2 - q^2\), \(b = 2pq\), \(c = p^2 + q^2\). Example row 1: \(x = 12/5 = 2.24\), giving triple (119, 120, 169). Sexagesimal base: \(60 = 2^2 \cdot 3 \cdot 5\), highly composite, enabling exact reciprocals for regular numbers (5-smooth numbers: \(2^a \cdot 3^b \cdot 5^c\)) | CONNECTION: The reciprocal pair method is structurally identical to the Euclidean formula for primitive triples: \((m^2-n^2, 2mn, m^2+n^2)\). The sexagesimal system's preference for regular numbers (whose reciprocals terminate) creates a natural lattice of 5-smooth numbers — this is the multipli Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com