Plimpton 322: Pythagorean Triples from Regular Sexagesimal Reciprocal Pairs — E8 Intelligence Research

FINDING: Plimpton 322 encodes 15 Pythagorean triples generated by a systematic regular sexagesimal reciprocal-pair (p,q) algorithm, not by trial-and-error or a single formula. MATH: - Core generator: for regular sexagesimal pair (p, q) with p > q > 0, both p and q having only prime factors 2, 3, 5 (i.e., 60-regular), form: - short leg: a = p² − q² - long leg: b = 2pq - diagonal: c = p² + q² - Normalized ratio on tablet is (b/a)² or equivalently (c/a)² = 1 + (b/a)², tabulated as sexagesimal values. - Column I values are (c/a)² = (p²+q²)² / (p²−q²)², ranging from ~1.463 to ~1.984 (sexagesimal 1;27,54,50,10 down to 1;58,54,02,30). - Bruins reconstruction: p, q drawn from standard reciprocal table (2·3ᵃ·5ᵇ), yielding the "missing pairs" gap that the arXiv:1109.3814 modification addresses by a natural cultural constraint. - Sexagesimal base-60 = 2²·3·5, so regularity (terminating reciprocals) requires p, q ∈ {2^a 3^b 5^c}. CONNECTION: - Base-60 regularity is a lattice condition: the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com