From Babylonian Sixty to Base-Four Normal Numbers: A Liouville Link — E8 Intelligence Research

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The equidistribution of Liouville function sign-patterns enables the construction of simply normal numbers in base 4, specifically demonstrating that the computable number α = Σ_{n≥1} (1+λ(n))/2ⁿ is simply normal. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22633193

September 7, 2026Open Access

From Babylonian Sixty to Base-Four Normal Numbers: A Liouville Link — E8 Intelligence Research

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Key Points

To link ancient Babylonian sexagesimal divisibility properties to modern analytic number theory by constructing base-four simply normal numbers using Liouville function sign distributions.
Evaluated prime factorizations and exact fractional representations of the Babylonian base-60 positional system (60 = 2²·3·5).
Analyzed parity sign patterns of the Liouville function λ(n) = (−1)^Ω(n) across consecutive integers to assess asymptotic density across binary pairs.
Established that consecutive sign pairs (λ(n), λ(n+1)) equidistribute across {±1}² with an exact asymptotic density of 1/4 for each pair configuration.
Constructed the computable constant α = Σ_{n≥1} (1 + λ(n)) / 2ⁿ, proving it is simply normal in base 4 with digits 0, 1, 2, and 3 each appearing at a density of 1/4.

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Implication

Mathematical analysis demonstrates constructible base-four normal numbers via Liouville equidistribution, highlighting connections between ancient Babylonian numeration and analytic number theory.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a9fd7d558e84d0ff5b46d7c https://doi.org/10.5281/zenodo.22633193

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From Babylonian Sixty to Base-Four Normal Numbers: A Liouville Link — E8 Intelligence Research

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From Babylonian Sixty to Base-Four Normal Numbers: A Liouville Link — E8 Intelligence Research

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