Structure of Triangular Numbers Modulo <i>m</i>

Abstract

This work focuses on the structure and properties of the triangular numbers modulo m. The most important aspect of the structure of these numbers is their periodic nature. It is proven that the triangular numbers modulo m forms a 2m-cycle for any m. 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 m has self-similarity and scaling features which are discussed as well.

Keywords

• triangular numbers
• periodic sequences
• self-similarity
• scaling