Babylonian Sieve Algorithm for Constructing Sexagesimal Reciprocal Tables — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Babylonian reciprocal tables were constructed via an algorithmic sieve over regular numbers (2^a·3^b·5^c·…), not brute force; the arxiv paper (1401.0065) details how large tables were generated by iterative multiplication by 2, 3, 5 — the prime factors of 60. | MATH: Regular number n = 2^a·3^b·5^c (a,b,c ≥ 0). Reciprocal 1/n has finite sexagesimal expansion iff n divides 60^k for some k. Table construction: start with 1, repeatedly multiply by {2,3,5}, sort, truncate at a bound (e.g., n ≤ 60^m). Reciprocal pairs (n, 1/n) satisfy n·(1/n) = 1 exactly in base-60. No closed-form for 1/223 or 1/239 — these are *irregular* (223, 239 are primes > 5), so they do NOT appear in standard tables; their reciprocals are infinite repeating sexagesimals. | CONNECTION: The set of regular numbers forms a multiplicative lattice in ℤ³ (exponents a,b,c). The ratio structure mirrors the 3D root lattice A₃ (fcc) — the fundamental parallelepiped has volume log(2)·log(3)·log(5). The density of regular
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026babylonian,
title = {{Babylonian Sieve Algorithm for Constructing Sexagesimal Reciprocal Tables — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23447156},
url = {https://zenodo.org/records/23447156},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Babylonian Sieve Algorithm for Constructing Sexagesimal Reciprocal Tables — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/18
PB - Zenodo
DO - 10.5281/zenodo.23447156
UR - https://zenodo.org/records/23447156
AB - FINDING: Babylonian reciprocal tables were constructed via an algorithmic sieve over regular numbers (2^a·3^b·5^c·…), not brute force; the arxiv paper (1401.0065) details how large tables were generated by iterative multiplication by 2, 3, 5 — the prime factors of 60. | MATH: Regular number n = 2^a·3^b·5^c (a,b,c ≥ 0). Reciprocal 1/n has finite sexagesimal expansion iff n divides 60^k for some k. Table construction: start with 1, repeatedly multiply by {2,3,5}, sort, truncate at a bound (e.g., n ≤ 60^m). Reciprocal pairs (n, 1/n) satisfy n·(1/n) = 1 exactly in base-60. No closed-form for 1/223 or 1/239 — these are *irregular* (223, 239 are primes > 5), so they do NOT appear in standard tables; their reciprocals are infinite repeating sexagesimals. | CONNECTION: The set of regular numbers forms a multiplicative lattice in ℤ³ (exponents a,b,c). The ratio structure mirrors the 3D root lattice A₃ (fcc) — the fundamental parallelepiped has volume log(2)·log(3)·log(5). The density of regular Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -