Formal Mathematical Framework of a Modular Reduction Process via Digital Root: A Technical Revision (V2)
Fuente:
Zenodo
Saved in:
| Main Author: | |
|---|---|
| Format: | Recurso digital |
| Language: | English |
| Published: |
Zenodo
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866902181287821312 |
|---|---|
| author | Esposito, Andrea |
| author_facet | Esposito, Andrea |
| contents | <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> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19148443 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Formal Mathematical Framework of a Modular Reduction Process via Digital Root: A Technical Revision (V2) Esposito, Andrea digital root modular arithmetic number theory arithmetic functions modular reduction asymptotic density nilpotent function Harshad numbers <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> |
| title | Formal Mathematical Framework of a Modular Reduction Process via Digital Root: A Technical Revision (V2) |
| topic | digital root modular arithmetic number theory arithmetic functions modular reduction asymptotic density nilpotent function Harshad numbers |
| url | https://doi.org/10.5281/zenodo.19148443 |