Babylonian Sieve Algorithm for Constructing Sexagesimal Reciprocal Tables — E8 Intelligence Research

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