The Theory of Integer Classification, Distribution and Factorization

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 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 Factor-2 Theorem) is established that allows for the development of the compressed factorization space using a concept of self-testing 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