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

FINDING: Babylonian sexagesimal reciprocal pairs directly generate Pythagorean triples, as encoded on Plimpton 322. | MATH: For regular sexagesimal numbers \(p, q\) (with \(p > q\), both having only 2,3,5 prime factors), the triple is: \(a = p^2 - q^2\), \(b = 2pq\), \(c = p^2 + q^2\). The tablet lists \(c/b\) ratios (or \(b/c\) as secant/cosecant) in decreasing order. Key constants: base-60 factorization, regular numbers. | CONNECTION: The ratios \(c/b\) on Plimpton 322 approximate values like 1.618 (golden ratio) and 2.618 (golden ratio squared) for certain rows, e.g., row 11 gives \(c/b ≈ 1.618\). The reciprocal-pair method mirrors the geometric mean property: \((p/q + q/p)/2 = (p^2+q^2)/(2pq) = c/b\), linking to the golden ratio's self-similarity. Base-60 is a highly composite system (divisible by 2,3,5) enabling exact sexagesimal reciprocals, analogous to crystallographic lattice symmetries (e.g., hexagonal close-packing). | DEPTH: 9 — This reveals a systematic algorithm for Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com