From Vortex Mathematics to Smith Numbers: Demystifying Number Structures and Establishing Sieves Using Digital Root

Abstract

Proficiency in number structures depends on a continuous development and blending of intricate combinations of different types of numbers and its related characteristics. The purpose of this paper is to unpack the mechanisms and underlying notions that elucidate the potential process of number construction and its inherent structures. By employing the concept of digital root, we show how juxtaposed assumptions can play in delineating generalized models of number structures bridging the abstract, the numerical, and the physical worlds. While there are numerous proposed ways of constructing Smith numbers, developing a generalized algorithm could help provide a unified approach to generating number structures with inherent commonalities. In this paper, we devise a sieve for all Smith numbers as well as other related numbers. The sieve works on the principle of digital roots of both S (N), the sum of the digits of a number N and that of  S (N), the sum of the digits of the extended prime divisors of N. Starting with S (N) =  S (p.q.r…), where p, q, r,…, are the prime divisors whose product yields N and whose digital root (n) equals to that of S (N) thus S (N) = n + 9x; x є N. The sieve works on finding the proper value of x that renders a Smith number N. In addition to the sieve, new related numbers could emerge.