Babylonian Base-60: Regular Numbers and Pythagorean Triples in Plimpton 322 — E8 Intelligence Research

FINDING: Old Babylonian base-60 system enabled efficient generation of regular numbers and primitive Pythagorean triples, with Plimpton 322 revealing advanced trigonometric or ratio-based computation. MATH: - Base-60 (sexagesimal) place-value system: digits 0–59, positional notation. - Regular numbers: integers of form \(2^a 3^b 5^c\) (a,b,c integers ≥0), reciprocals have finite sexagesimal expansions. - Primitive Pythagorean triple generation (Dickson method): find integers \(r,s,t\) such that \(r^2 = 2st\), then triple = \((r+s, r+t, r+s+t)\). Equivalent to Euclid's formula: \(a = m^2 - n^2\), \(b = 2mn\), \(c = m^2 + n^2\) with \(m>n\), coprime, opposite parity. - Plimpton 322 table: columns of \((b^2, a, c)\) or \((b/a, c/a)\) for 15 triples, likely computed via regular numbers and reciprocals. CONNECTION: - Base-60 directly links to geometric harmony ratios: 1/60 = 0.01666..., but more critically, regular numbers produce ratios like 0.618 (≈ 1/φ) via \(2^{-1}3^{-1}5^ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com