The SOM Numbers (part I)

Abstract: This study delves into the interconnection between square and oblong numbers and their relation to number theory, significantly enhancing the understanding of numerical structure and prime number distribution. Initially, we examine the interweaving pattern between square and oblong numbers, revealing a periodic classification of integers and establishing an analogy with the interlocked teeth of a zipper. This analysis lays the foundation for an in-depth revision of Legendre's Conjecture, offering new insights into the distribution of prime numbers along the numerical line and uncovering previously unexplored mathematical connections.

We advance in the field of quadratic sequences by introducing what we term the "3 Basic Theorems of Quadratics." These theorems are fundamental for understanding the properties and behaviors of quadratic functions, including the application of Taylor shift and offset.

We develop a comprehensive atlas mapping all sequences of the type "quadratic minus an element of the quadratic," providing extensive documentation and categorization of these sequences. This atlas allows for the exploration of a wide range of behaviors and patterns in quadratic functions. Furthermore, we demonstrate how finite sequences, specifically trios, can uniquely characterize each of the infinite quadratic sequences.

Complementing the atlas, we create a universe of matrices of quadratic sequences, logically and structurally organizing all sequences. These matrices illuminate patterns, symmetries, and recurring properties in quadratic sequences, offering a powerful tool for solving open mathematical problems. The ultimate goal is to establish a definitive system for organizing all connections between quadratic sequences, fostering significant advancements in various areas of mathematics, such as number theory, algebra, and their practical applications.

Please note that this document is a preprint and has not yet been peer-reviewed. It is intended for discussion and feedback within the scientific community to refine the approaches and conclusions presented.

Keywords: Square Numbers, Oblong Numbers, Number Theory, Quadratic Sequences, Mathematical Matrices.

2020 Mathematics Subject Classification: 11A41; 05A15; 11B37; 11C08.