The Distribution of Prime Numbers and Finding the Factor of Composite Numbers without Searching

In this paper, there are 5 sections of tables represented by 5 linear sequence functions. There are two one-variable sequence functions that they are able to represent all prime numbers. The first one helps the last one to produce another three two-variable linear sequence functions. With the help of these three two-variable sequence functions, the last one, one-variable sequence function, is able to set apart all prime numbers from composite numbers. The formula shows that there are infinitely many prime numbers by applying limit to infinity. The three two-variable sequence functions help us to find the factor of all composite numbers.


Introduction
A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.A natural number greater than 1 that is not a prime number is called a composite number.For example, 5 is prime because 1 and 5 are its only positive integer factors, whereas 6 is composite because it has the divisors 2 and 3 in addition to 1 and 6.The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering.The uniqueness in this theorem requires excluding 1 as a prime because one can include arbitrarily many instances of 1 in any factorization, e.g., 3, 1 × 3, 1 × 1 × 3, etc. are all valid factorizations of 3. The property of being prime (or not) is called primality.A simple but slow method of verifying the primality of a given number n is known as trial division.It consists of testing whether n is a multiple of any integer between 2 and n .Algorithms much more efficient than trial division have been devised to test the primality of large numbers.Particularly fast methods are available for numbers of special forms, such as Mersenne numbers.As of April 2014, the largest known prime number has 17,425,170 decimal digits.There are infinitely many primes, as demonstrated by Euclid around 300 BC.There is no known useful formula that sets apart all of the prime numbers from composites.However, the distribution of primes, that is to say, the statistical behaviour of primes in the large, can be modelled.The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or to the logarithm of n.Many questions regarding prime numbers remain open, such as Goldbach's conjecture (that every even integer greater than 2 can be expressed as the sum of two primes), and the twin prime conjecture (that there are infinitely many pairs of primes whose difference is 2).Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers.Primes are used in several routines in information technology, such as public-key cryptography, which makes use of properties such as the difficulty of factoring large numbers into their prime factors.Prime numbers give rise to various generalizations in other mathematical domains, mainly algebra, such as prime elements and prime ideals [1].
In this paper I am going to prove that: 1) 1 is prime number, so the definition of prime number becomes "A prime number (or a prime) is a natural number greater than or equal to 1 that has no positive divisors other than 1 and itself." 2) There is useful formula that sets apart all of the prime numbers from composites or the distribution of primes.
3) Goldbach's conjecture and the twin prime conjecture.4) Finding the factor of any odd composite numbers without searching.For example to find the factor of composite number n = 114381625757888867669235779976146612010218296721242362562561842935706935245733897830597 123563958705058989075147599290026879543541 [2] the only thing we need to have is microsoft Excel software with big cell and a little bit big screen computer and track the number 105 2 n − from the two tables using two functions that we will discuss.My paper is important to find the factor composite numbers and prime numbers as we want as fast as possible and faster than trial division and algorithmic methods and perfect.

Preliminary
Definition (When Is a Number Divisible by 3) [3].
If the sum of the digits of a number is divisible by 3 , then the original number is divisible by 3. Definition (When Is a Number Divisible by 5) [3].
If the last digit of the number being inspected for divisibility is either a 0 or 5, then the number itself will be divisible by 5.
Definition (When Is a Number Divisible by 7) [3].Delete the last digit from the given number and then subtract twice this deleted digit from the remaining number.If the result is divisible by 7, the original number is divisible by 7.This process may be repeated if the result is too large for simple inspection of divisibility of 7.
Definition (The th n term of Arithmetic sequence).The th n term of an Arithmetic sequence is given by ( ) , where 1 a is the first term of the sequence and d is the common difference of the sequence.

Tables
)( )   S be disjoint subsets of the set of natural numbers whose union is equal to the set of natural numbers, then there exists 1 p and 2 p such that ( )( )  ( )

(Twin prime theorem)
There are infinitely many pairs of primes whose difference is 2.
Proof.Suppose there are infinitely many numbers ( ) ( ) , and ( ) + , where ( ) ( ) and ( ) ( ) ) p ≥ is the given th n term prime number, then n can be calculated as: ( , n f p p = number of elements of 1 f less than p, ( , n f p p = number of elements of 2 f less than p and ( ) , n f p p = number of elements of 3 f less than p.
Proof.Since ( ) 105 2 g p p = + is prime number for 1 p ≥ and 1 2 3 ,5 ,7 , , , p n n n f f ≠ and 3 f .Now let 1 2 3 4 5 , , , , , n n n n n and 6 n be in the set of natural numbers such that ( ) n ∈ℵ is the first term less than p, this implies that 3 1 3 k n = .Thus we have 3  3 k number of terms which are multiples of 3.
( ) and Thus every even integer greater than or equal to 2 can be expressed as the sum of two primes.□ Theorem 17.

Table 1 .
My magical table.

Theorem 16. (Goldbach's theorem)
Every even integer greater than or equal to 2 can be expressed as the sum of two primes.
the set of all odd composite numbers.□ Theorem 14.The sequence function, ( ) 105 2 1 g f or ( ) 2 g f or ( ) 3 g f or ( )