Babylonian Reciprocals and Plimpton 322: Generating Pythagorean Triples in Base-60 — E8 Intelligence Research

FINDING: Babylonian reciprocal-pair algorithm generates Pythagorean triples via sexagesimal regular numbers, with Plimpton 322 encoding exact fractional ratios (not just triples) in base-60. MATH: - Core algorithm: For regular numbers \( p, q \) (sexagesimal reciprocals, \( pq = 60^k \)), set \( a = p - q \), \( b = 2 \), \( c = p + q \) (scaled) → yields Pythagorean triple \( (a, b, c) \) with \( a^2 + b^2 = c^2 \). - Plimpton 322 lists 15 rows of \( ( short side, diagonal ) \) with \( long side = 1 \) (normalized), giving ratios \( short/long = p^2 - q^2/2pq \) and \( diagonal/long = p^2 + q^2/2pq \). - Sexagesimal regular numbers: \( p, q \) of form \( 2^a 3^b 5^c \) (and their reciprocals), enabling exact finite base-60 fractions. - Key constants: \( 60 = 2^2 · 3 · 5 \); reciprocal pairs like \( (2, 30), (3, 20), (4, 15), (5, 12), (6, 10) \) in base-60. CONNECTION: - The ratio \ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com