The Theory of Integer Classification, Distribution and Factorization 

Author:
 Mbulawa, Zwide 
ISBN:  9781483630861 
Publication Date:  Aug 2013 
Publisher:  Xlibris Corporation LLC

Book Format:  Paperback 
List Price:  USD $81.99 
Book Description:

The Theory of Integer Classification, Distribution and Factorization The approach applied here is a unified classification system that uses an algebraic framework to establish a relationship and distribution system for integers. All numbers are defined algebraically through the universal number distribution theorem (UNDT). The advantage of the system is that: 1. It explains the gap theory of prime numbers, and the source of prime number randomness. 2. It revises and expands the concept...
More DescriptionThe Theory of Integer Classification, Distribution and Factorization The approach applied here is a unified classification system that uses an algebraic framework to establish a relationship and distribution system for integers. All numbers are defined algebraically through the universal number distribution theorem (UNDT). The advantage of the system is that: 1. It explains the gap theory of prime numbers, and the source of prime number randomness. 2. It revises and expands the concept and definition of a prime number. 3. It derives several primality test algorithms, and sieve structures. The theory has more than five ways of accurately testing for primality. 4. It derives the useful Integer Product Law that governs the distribution of the product between integers. It forms the basis for the development of the factorization engines. 5. It derives a general prime number factorization algorithm the structured factorization method (SFM framework). That is, factorization is precise and bounded in a given space. 6. It derives an algorithm that does both factorization and primality testing simultaneously (failure test approach). 7. It creates a factorization engine that runs on a transient database that is unique for a given number. This is the "6" factorization engine. This reduces time in looking for prime factors, and makes it not necessary to refer to a database of prime numbers when factorizing. 8. A powerful theorem (The Factor2 Theorem) is established that allows for the development of the compressed factorization space using a concept of selftesting primality. 9. For the compressed factorization space, another factorization engine is defined leading to further compression of the factorization space. This is the "12" engine that is extremely efficient in creating the transient database for prime factors (turbo charged factorization). From classification, without a computer program but manually through an excel worksheet, the system was used to find factors of this number that was chosen at random. 100 785 423 193 771 = 3²x103x108 722 139 373 This was done in about ten minutes. The power of the classification system lies in the fact that there is no need for a database prime numbers since through the classification every number has a unique coordinate reference. Classification also yields the natural primality testing method, where through classification without any further computation, this 69 digit number is definitely not a prime number. You can show this in less than a minute! 789900541289753188975009123215467980768572971397525750310229845601223 The theory also defines the Merge operator that conceptualizes predictive factorization. Knowing the prime factors of one number, you can correctly predict two or more factors of a much larger random number without any further calculation. Through the same operator, you can also predict that one number will definitely have more prime factors than the other. Interesting and unusual! The book suggests new approaches in the understanding of integers in terms of classification, distribution and factorization. It is referred to as a unifying theory because it covers all the three areas. The book also provides a unique and clear understanding of the definition, behavior and distribution of prime numbers through the classification system. This leads to a revision of the Euclidean premise of infinity in regard to prime number distribution and the count function.