Plimpton 322: A Systematic Sexagesimal Algorithm for Pythagorean Triples — E8 Intelligence Research

Abstract

FINDING: Plimpton 322 encodes a systematic sexagesimal algorithm for generating Pythagorean triples via reciprocal pairs of regular numbers, not a random table. | MATH: For regular numbers \(p, q\) (with \(q < p\), both having only prime factors 2, 3, 5), the triple is \((a, b, c) = (p^2 - q^2, 2pq, p^2 + q^2)\). The tablet's columns correspond to \(\frac{p}{q}\) and \(\frac{q}{p}\) as sexagesimal reciprocals, yielding ratios \(\frac{a}{c}\) and \(\frac{b}{c}\) that are exact in base-60. The "missing pairs" in Bruins' reconstruction are explained by a modified rule: the algorithm generates all triples where the hypotenuse \(c\) is a regular number (i.e., \(c = p^2 + q^2\) has only 2, 3, 5 as prime factors), which is a stricter condition than merely \(p, q\) regular. | CONNECTION: The regular numbers form a multiplicative lattice in \(\mathbb{Q}^+\) generated by \(\{2, 3, 5\}\) — a rank-3 lattice. The reciprocal pairs \((p, q)\) and \((q, p)\) correspond to points on the unit circle in Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com