From Babylonian Sixty to Base-Four Normal Numbers: A Liouville Link — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Babylonian base-60 numeration underpins ancient astronomy; modern result links Liouville function sign-pattern equidistribution to constructible simply normal numbers in base 4. MATH: - Base-60: positional system with 60 = 2²·3·5; divisors {1,2,3,4,5,6,10,12,15,20,30,60} → exact fractions for 1/2, 1/3, 1/4, 1/5, 1/6, 1/10, 1/12, 1/15, 1/20, 1/30, 1/60. - Liouville function λ(n) = (−1)^Ω(n) (Ω = total prime factors with multiplicity). - Key theorem: double sign patterns (λ(n), λ(n+1)) equidistribute — i.e., for each (ε₁,ε₂) ∈ {±1}², density = 1/4. - Application: α = Σ_{n≥1} (1+λ(n))/2ⁿ is simply normal in base 4 (each digit 0,1,2,3 appears with density 1/4). - Note: α is computable (since λ is computable), yet its base-4 normality is non-trivial. CONNECTION: - Base-60's rich divisibility yields ratios 1/60, 1/30, 1/20, 1/15, 1/12, 1/10, 1/6, 1/5, 1/4, 1/3, 1/2 — these include 0.2, 0.25, 0.333…, 0.5, but NOT 0.382, 0.618, 0.786, 1.618, 2.618 (those are irrational Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7211866004
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