Factorial Digit Occurences

\begin{abstract}

In this paper, we present a comprehensive study of the occurrences of digits in the factorials of prime numbers when these factorials are represented in various bases. We explore the mathematical properties and significance of factorials and their digit distributions, demonstrating their relevance to the components that build up Euler's number \( e \). Our approach involves several computational techniques, including factorial calculation, base conversion, and digit counting. Furthermore, we develop an interactive Python program that visualizes these digit occurrences in a three-dimensional space, providing a dynamic tool for analyzing the behavior of factorials of prime numbers across different bases. This visualization tool leverages the capabilities of symbolic computation libraries and interactive widgets, enabling users to intuitively explore how digit frequencies evolve with changes in base and range of prime numbers. Our findings not only deepen the understanding of factorial properties but also offer insights into the intricate nature of mathematical constants and their computational representations.

\end{abstract}