Structure of Triangular Numbers Modulo m
Abstract
:1. Introduction
2. Information and Data Compression
2.1. Sets and Sequences
2.2. Information Content of Sets
3. Main Structure Theorems
3.1. Main Cycle Structure Theorem
3.2. Symmetry Theorems
4. Specific Structure Theorems
4.1. Theorems Related to
4.2. Theorems Involving Cases When m Is Prime
4.3. Theorems Involving Cases When m Is a Prime Squared
5. Property Theorems
6. Saturation Theorems
7. Special Sets
7.1. Monoids and Reduced Monoids
7.2. Perfect Squares and Nonsquares
8. Graph Representation
8.1. Construction of the Graph Representation
8.2. Structure of the Graph Representation
8.3. Perfect Squares and Nonsquares in the Graph Representation
9. Self-Similarity
10. Scaling Properties
10.1. Sprays
10.2. Sprays, Renormalization, and Fractal Character
11. Application to Harmonic Analysis
12. Conclusions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Ulness, D.J. Structure of Triangular Numbers Modulo m. AppliedMath 2022, 2, 326-358. https://doi.org/10.3390/appliedmath2030020
Ulness DJ. Structure of Triangular Numbers Modulo m. AppliedMath. 2022; 2(3):326-358. https://doi.org/10.3390/appliedmath2030020
Chicago/Turabian StyleUlness, Darin J. 2022. "Structure of Triangular Numbers Modulo m" AppliedMath 2, no. 3: 326-358. https://doi.org/10.3390/appliedmath2030020