Prime Sieve and Factorization Using Multiplication Table

Using the properties of the table sieve, we can determine whether all given number, positive integer G, is a prime and whether it is possible to factor it out.


Introduction
The sieve of Eratosthenes was successful in filtering out composite numbers using the fact that it is easy to calculate multiples but it was not successful to find the relationship between filtered-out composite numbers.Our approach method has found a way to filter out all the primes using table multiplication.We could effectively find a factor of a given composite number.

Generating the Composite Number of the 12n+1, 5, 7, 11 Series
Every prime number except 2 and 3 is contained in the 12n+1, 5, 7, 11 series, is sorted into 4 kinds of remainder groups 1, 5, 7, and 11 and belongs to at least one of these 4 groups.Let us denote the set A n is all elements of the 12n+1, 5, 7, 11 series (n=0, 1, 2); the set P n is all elements of prime numbers as comprised in A n ; the set C n is all elements of composite numbers as comprised in A n .

Algorithmic Description
2.1.1All Prime Numbers but 2 and 3 Exist in Forms of 12n+1, 5, 7, 11 with a Period of 12n Proof.
(i) All natural numbers can be represented with a period of 12.
(ii) All even numbers but 2 are not prime numbers.
(iv) 12n+9 is not a prime number.
As a results, every prime number but 2 and 3 is contained in the periodic n A n .So, let us denote this series as follows: www.ccsenet.org/jmr

All Elements of the
Therefore, C n ≥ 25.Additionally, it is possible to find all values of n C n of the 12n+1, 5, 7, 11 series in the results of a matrix-multiplication of A n × A n that are greater than 5.

The Structure of a Matrix-multiplication of A n × A n
Unlike prime numbers, which are unpredictable, the composite numbers are formed by sixteen arithmetic progression groups.This means that composite numbers in principle are predictable because whole composite numbers follow this rule.However, the composite numbers are made up of sixteen arithmetic progressions and it is difficult to see the whole of the arithmetic progressions, whose number increases, without necessary computations and information media that can store the computed results.If you can find the computed results of various arithmetic progressions intuitively, you can predict the rule that governs the composite numbers.This immediately means that you will be able to find the rule that governs the prime numbers.It is not a problem of whether or not the composite numbers are predictable but a problem of human perception.
See Table 1.However, the commutative law does not hold if α β.So, depending on the orders of α and β, the results are different for the diagonal elements.An asymmetric table has twelve cases.

Substitution
We have seen that all the composite numbers but 2 and 3 can be represented in a form of (12x+α)(12y+β).Then, how can we determine if a given positive integer, G, is prime or composite?
Let G be an arbitrary positive integer.
(i) Check if G is a multiple of 2 or 3.If G is a multiple of one of these, it is a composite number.
(ii) If G is not a multiple of 2 or 3, G' s remainder R when divided by 12 is R ∈ 1, 5, 7, 11 and R is the a number in the table multiplication elements, then G is a composite number.If R is not the same as any of the multiplication elements, then G is a prime number.
If a given number, G, is not a multiple of 2 or 3, we can express it as follows.
If we substitute xy, βx + αy with X and Y, respectively, The result is the following.

Determination of a Valid Domain
Let us apply the arithmetic mean and geometric mean to xy and βx + αy.From βx + αy=Y and xy = X, When x = 1, X (xy) is the minimum.Therefore, the minimum of X (xy) is The maximum of X (xy) is Vol. 4, No. 3;2012 (i) If x + y is even, then the maximum is achieved when x = y.
(ii) If x + y is odd, then the maximum is achieved when x + 1 = y.
-If the maximum and minimum of X(xy) is not an integer, then we can make it an integer by rounding it up.
If we can determine the valid domain, we can make the following table list of (X, Y).

Finding Factors: First Method
x, n, r is positive integer (x, r contatins zero), Since Since x is zero or a positive integer, we can find integer roots (n, r) from If an integer root, (n, r), exists, then we can find integer (x, y).
increases by 1 and Y decreases by 12. So, X and Y have properties of an arithmetic progression.Let two arithmetic progressions, X and Y, be X = n + a and Y = −12n + b, respectively.(X, Y) Tables pairs Is there an efficient method to find a valid set of (X, Y), which has an integer root, from (a 1 , b 2 ), (a 2 , b 2 ), (a 3 , b 3 ), (a 4 , b 4 ), (a 5 , b 5 ), (a 6 , b 6 ),..., (a n , b n ).
a n Y b 1 b 2 b 3 b 4 b 5 b 6 b 7 b 8 b 9 b 1 0 ... b n

Table 1 .
The result list of multiplication table

Table 2 .
Finding integer values of x and y X a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 1 0 ...