Plimpton 322: Sexagesimal Reciprocal-Pair Algorithm for Systematic Pythagorean Triples — E8 Intelligence Research

FINDING: Plimpton 322 encodes a systematic set of Pythagorean triples in base-60, generated by a reciprocal-pair algorithm, with missing pairs explained by a natural sexagesimal accounting scheme. MATH: For a regular sexagesimal number \( p \) (i.e., \( p = 2^a 3^b 5^c \)), let \( q = 1/p \) in base-60. Then the triple is \((a, b, c)\) with \( a = p - q \), \( b = 2 \), \( c = p + q \), scaled to integer sides. The tablet lists 15 such triples, with \( p \) ranging from 2;24 to 1;48 (sexagesimal). Key constants: 1;24,51,10 (approximation of √2) and 0;30,0,0 (1/2). CONNECTION: The reciprocal-pair method relies on regular numbers (divisors of 60), which form a multiplicative lattice isomorphic to \( \mathbb{Z}^3 \) under exponents of 2,3,5. This lattice has 6-fold rotational symmetry in the plane of exponents (since 2,3,5 are primes, but the group of units modulo 60 is cyclic of order 8, not 6). However, the base-60 system itself is 6-fold in the sense that 60 = 2²·3·5, and the regul Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com