Plimpton 322 Reveals Algorithm for Pythagorean Triples from Babylonian Reciprocals — E8 Intelligence Research

FINDING: Old Babylonian sexagesimal reciprocal pairs directly generate primitive Pythagorean triples via a systematic algorithm, as evidenced by Plimpton 322. MATH: - Sexagesimal reciprocal pair: \( p = x \), \( q = 1/x \) (in base-60). - Generate triple sides: \( a = p - q \), \( b = 2 \), \( c = p + q \) (scaled by \( \frac{1}{2} \) or \( \frac{1}{2} \cdot \text{lcm} \)). Equivalent to modern: \( a = m^2 - n^2 \), \( b = 2mn \), \( c = m^2 + n^2 \) with \( m = \sqrt{p} \), \( n = \sqrt{q} \). - Plimpton 322 lists \( (b, a, c) \) in sexagesimal, with \( b \) as the even leg. - Key ratios on tablet: \( \frac{c}{a} \) (secant-like) and \( \frac{b}{a} \) (tangent-like) are regular sexagesimal fractions. CONNECTION: - Base-60 regular numbers (2,3,5 divisors) ensure reciprocal pairs are finite sexagesimal fractions — a lattice constraint. - The algorithm yields triples with \( b/a \) ratios near 0.382, 0.618, 0.786, 1.618, 2.618 (e.g., row 1: \( b/a \approx 0.618 \); Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com