The Chairux Divisibility Transformation: A Systematic Approach to Prime Divisibility Testing

Abstract

The Chairux divisibility transformation is a novel method for checking the divisibility of a number Q by a prime number p. Unlike traditional divisibility rules, which are limited to small primes, the Chairux transformation provides a systematic approach for testing divisibility by any prime. This paper presents the mathematical foundations of the Chairux transformation, derives its key properties, and demonstrates its applications in number theory and cryptography. We also introduce three methods for finding the prime divisor N given the Chairux coefficients (α, β): the structural method, the iteration method, and the bi-section method. The paper concludes with examples and case studies illustrating the practical utility of the transformation.