Structure of Triangular Numbers Modulo <i>m</i>
This work focuses on the structure and properties of the triangular numbers modulo <i>m</i>. The most important aspect of the structure of these numbers is their periodic nature. It is proven that the triangular numbers modulo <i>m</i> forms a <inline-formula><math x...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-07-01
|
| Series: | AppliedMath |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2673-9909/2/3/20 |
| _version_ | 1797466023756038144 |
|---|---|
| author | Darin J. Ulness |
| author_facet | Darin J. Ulness |
| author_sort | Darin J. Ulness |
| collection | DOAJ |
| description | This work focuses on the structure and properties of the triangular numbers modulo <i>m</i>. The most important aspect of the structure of these numbers is their periodic nature. It is proven that the triangular numbers modulo <i>m</i> forms a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>m</mi></mrow></semantics></math></inline-formula>-cycle for any <i>m</i>. Additional structural features and properties of this system are presented and discussed. This discussion is aided by various representations of these sequences, such as network graphs, and through discrete Fourier transformation. The concept of saturation is developed and explored, as are monoid sets and the roles of perfect squares and nonsquares. The triangular numbers modulo <i>m</i> has self-similarity and scaling features which are discussed as well. |
| first_indexed | 2024-03-09T18:30:00Z |
| format | Article |
| id | doaj.art-17609f1a120646a080d4926b7aaaf703 |
| institution | Directory Open Access Journal |
| issn | 2673-9909 |
| language | English |
| last_indexed | 2024-03-09T18:30:00Z |
| publishDate | 2022-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | AppliedMath |
| spelling | doaj.art-17609f1a120646a080d4926b7aaaf7032023-11-24T07:33:03ZengMDPI AGAppliedMath2673-99092022-07-012332635810.3390/appliedmath2030020Structure of Triangular Numbers Modulo <i>m</i>Darin J. Ulness0Department of Chemistry, Concordia College, Moorhead, MN 56562, USAThis work focuses on the structure and properties of the triangular numbers modulo <i>m</i>. The most important aspect of the structure of these numbers is their periodic nature. It is proven that the triangular numbers modulo <i>m</i> forms a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>m</mi></mrow></semantics></math></inline-formula>-cycle for any <i>m</i>. Additional structural features and properties of this system are presented and discussed. This discussion is aided by various representations of these sequences, such as network graphs, and through discrete Fourier transformation. The concept of saturation is developed and explored, as are monoid sets and the roles of perfect squares and nonsquares. The triangular numbers modulo <i>m</i> has self-similarity and scaling features which are discussed as well.https://www.mdpi.com/2673-9909/2/3/20triangular numbersperiodic sequencesself-similarityscaling |
| spellingShingle | Darin J. Ulness Structure of Triangular Numbers Modulo <i>m</i> AppliedMath triangular numbers periodic sequences self-similarity scaling |
| title | Structure of Triangular Numbers Modulo <i>m</i> |
| title_full | Structure of Triangular Numbers Modulo <i>m</i> |
| title_fullStr | Structure of Triangular Numbers Modulo <i>m</i> |
| title_full_unstemmed | Structure of Triangular Numbers Modulo <i>m</i> |
| title_short | Structure of Triangular Numbers Modulo <i>m</i> |
| title_sort | structure of triangular numbers modulo i m i |
| topic | triangular numbers periodic sequences self-similarity scaling |
| url | https://www.mdpi.com/2673-9909/2/3/20 |
| work_keys_str_mv | AT darinjulness structureoftriangularnumbersmoduloimi |