Unit Fractions and the Erdã–s-Straus Conjecture

Shannon, A. G.

doi:10.24297/jam.v12i6.3848

Cited by 1 publication

(1 citation statement)

References 12 publications

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“…To develop a methodology to count cyclic object-relative uniform temporal elements, or units of a cycle (rotation, revolution, axial precession, and so on) each unit can be seen as a portion (k) of a whole (N). A unit fraction, described using ancient Egyptian interpretations 17 , cannot be converted into decimals or percentages; doing so would lose the connection to the modeled geometric part of a whole cycle. In this application, we have mixed unit fractions: the denominator (N) is the multiset multiplicity, the numerator (k) represents an ordered count of incremental elements (temporal units of a whole cycle), and the whole number is the count of completed cycles.…”

Section: Model Developmentmentioning

confidence: 99%

Dynamical astronomical object-relative cyclic signal inputs for discrete time modeling with calendric applications

Moore¹

2023

Preprint

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Accumulation of temporal errors occurs in continuous time modeling based on dynamical astronomical cycles; however, cyclic differences can also be advantageous. Dynamical cycles in nature, especially astronomical cycles, vary and lack uniformity. Unifying discrete time modeling with continuous time can chart these differences throughout linear time, building a reference repository. Ancient timekeeping systems are combined with modern computation and set theories, and when integrated with a signal system, cyclic fundamental temporal elements are created. Uniform scaling elements create unique and uniform cyclic and object-relative natural number units of time, each convertible to SI unit seconds. Despite countless Julian calendar-based comparison studies on timekeeping, ancient methodologies remain a mystery. This approach opens a hypothesized methodology that is testable for ancient observational calendric timekeeping systems. We present the first known pairing function to synchronize and assemble various independent object-relative observational-based astronomical cycles and their temporal elements. The model is applied to test a comparison between a previously proposed calendar model derived from Neolithic China and the Mesoamerican timekeeping divisional system. A 13-lunation calendar year built upon a 20-year cycle is introduced as the first new hypothesized astronomical cyclic pairing for Mesoamerica in nearly 500 years. Results validate a hypothesized link between Mesoamerica and ancient China. Combining mathematical and computational methodologies, our cyclic object-relative signal inputs from continuous time can yield new applications in astrometry and physics, as well as tools for archaeological research.

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Section: Model Developmentmentioning

confidence: 99%