Plimpton 322's Sexagesimal Ratios: A Reconstruction Algorithm for Missing Pairs — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Plimpton 322 encodes a systematic sequence of regular sexagesimal ratios (p/q) generating Pythagorean triples, with a reconstruction algorithm that accounts for "missing pairs" via a natural modification of Bruins' method. | MATH: The tablet lists 15 rows of (sexagesimal) values corresponding to triples (a, b, c) with a² + b² = c². Reconstruction uses regular numbers p, q in base 60 (i.e., p, q whose prime factors are only 2, 3, 5) to form ratios p/q. The standard parametrization: a = p² − q², b = 2pq, c = p² + q² (or scaled variants). Bruins' scheme: choose p, q from a regular sexagesimal grid; the "missing pairs" arise from gaps in the sequence where p/q falls outside the tablet's chosen interval (likely 1 < p/q < 1.5, with decreasing q). The arXiv paper (1109.3814v2) proposes a modified rule that predicts subsequent rows beyond row 15, implying a continuous, ordered algorithm — likely iterating p, q via multiplication by regular factors (e.g., 2, 3, 5) to maintain constant
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
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- Babylonian Tablet Plimpton 322: Ancient Sexagesimal Trigonometry Predates Greeks — E8 Intelligence Research
BibTeX
@misc{caldin2026plimpton,
title = {{Plimpton 322's Sexagesimal Ratios: A Reconstruction Algorithm for Missing Pairs — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23658281},
url = {https://zenodo.org/records/23658281},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Plimpton 322's Sexagesimal Ratios: A Reconstruction Algorithm for Missing Pairs — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23658281
UR - https://zenodo.org/records/23658281
AB - FINDING: Plimpton 322 encodes a systematic sequence of regular sexagesimal ratios (p/q) generating Pythagorean triples, with a reconstruction algorithm that accounts for "missing pairs" via a natural modification of Bruins' method. | MATH: The tablet lists 15 rows of (sexagesimal) values corresponding to triples (a, b, c) with a² + b² = c². Reconstruction uses regular numbers p, q in base 60 (i.e., p, q whose prime factors are only 2, 3, 5) to form ratios p/q. The standard parametrization: a = p² − q², b = 2pq, c = p² + q² (or scaled variants). Bruins' scheme: choose p, q from a regular sexagesimal grid; the "missing pairs" arise from gaps in the sequence where p/q falls outside the tablet's chosen interval (likely 1 < p/q < 1.5, with decreasing q). The arXiv paper (1109.3814v2) proposes a modified rule that predicts subsequent rows beyond row 15, implying a continuous, ordered algorithm — likely iterating p, q via multiplication by regular factors (e.g., 2, 3, 5) to maintain constant Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -