Unveiling the Structure of Cayley-Dickson Algebras: Zero Divisor Counting and Listing, and a Novel Sign Compression Scheme

Abstract

This paper presents a comprehensive computational and empirical investigation into the structure and properties of Cayley-Dickson algebras ($A_x$, dimension $2^x$). We establish and validate an explicit formula, $N(x) = \frac{(2^x - 2)(2^x - 4)(2^x - 8)}{16}$, which accurately enumerates a specific class of unique zero divisor pairs for $x \geq 4$. Complementing this quantitative result, a detailed structural analysis of the multiplication tables reveals a recursive decomposition into $8\times8$ blocks, reflecting octonion substructures. We introduce a novel ``block type'' classification (`x' or `y') based on an indicator element ($e_{(8k, 8k+1)}$), which determines zero divisor location and highlights recursive patterns. Conceptual frameworks like an ``Observed Pattern Multiplication Table'' and a sign compression scheme guided this work. A Python script is provided. While OPMT is formally proven and computationally validated, the other many findings are empirical conjectures requiring formal mathematical development, yet they offer a substantially deeper understanding of Cayley-Dickson algebras.