The visualize formulation of divisibility

Abstract

For the purposes of cryptography it is necessary to develop effective methods and algorithms: to check the simplicity of integers; to find large prime numbers; to factorize integers. This paper studies a generalized method for constructing algorithms to check the divisibility of integers by a given number b in various number systems by analyzing the sets of divisors of base (a given number b), prebase (number b – 1), and postbase (number b + 1). It is indicated that the rules for testing divisibility by a given number may have different complexity depending on the number system used. The paper introduces the formulations of some theorems with proofs. The theorems are supported by concrete examples. These theorems can formulate for many divisibility rules for any number over the any base. Some numbers are although difficult over base 10, they are easy over another base. Some numbers, such as primes, have direct rules, but some composites have combined rules