5-Smooth Regular Numbers Underlie Babylonian Base-60 Exact Division — E8 Intelligence Research

FINDING: Babylonian sexagesimal (base-60) arithmetic is built on 5-smooth "regular numbers" (2^a·3^b·5^c), which possess finite reciprocals and enable exact division without approximation. MATH: Base-60; regular numbers n = 2^a·3^b·5^c; reciprocal 1/n terminates in sexagesimal iff n is 5-smooth. Plimpton 322 (OB, ~1800 BCE) encodes integer pairs (a,b,c) satisfying a² + b² = c² (Pythagorean triples), tabulated via the reciprocal-pair method: for regular x, the triple is generated from (x − 1/x)/2, (x + 1/x)/2, (x + 1/x)/2. Standard OB reciprocal table: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 1:00, 1:04, 1:12, 1:15, 1:20, 1:21, 1:30. CONNECTION: 60 = 2²·3·5 is the smallest number divisible by 1–6; its divisors {1,2,3,4,5,6,10,12,15,20,30,60} form a highly composite lattice. The regular-number set is closed under multiplication — a multiplicative monoid. Plimpton 322's triples derive from the same reciprocal structure that underlies the gol Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com