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| Hlavní autor: | |
|---|---|
| Médium: | Recurso digital |
| Jazyk: | angličtina |
| Vydáno: |
Zenodo
2026
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| Témata: | |
| On-line přístup: | https://doi.org/10.5281/zenodo.19148443 |
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- <p>This technical revision (V2) formalizes an arithmetic function defined on the set of positive natural numbers based on the concept of the digital root.</p> <p>The function F(n) = n (mod dr(n)) is rigorously analyzed, providing formal proofs of its structural properties. In particular, the work demonstrates the restriction of the range, including the strict exclusion of the value 8 in base 10, and proves that the function is nilpotent of index 2, meaning that F(F(n)) = 0 for all n.</p> <p>The asymptotic distribution of the function values is also derived, showing that the set of numbers satisfying F(n) = 0 (referred to as 9-Harshad numbers) has a natural density of approximately 52.42%.</p> <p>Finally, the framework is generalized to arbitrary positional numeral systems, proving that the exclusion of the value b − 2 is a universal structural property.</p> <p>Note: The core idea and underlying mechanism presented in this work were developed independently by the author. The mathematical formalization was refined with the support of assistive tools.</p>