Prime Number in number theory learning

Prime Number in number theory learning

1. Definitions:

Division: Set A and B to two integers, and B! = 0 if there is an integer c that makes a = BC, it is said that a is divided by B, or B is divided by a, it is recorded

B |;

Division with remainder:

A = QB + R;

This formula is called the remainder of the records with Division R = a mod B;

2. Division nature

1. If a | B and A | C, there is a | bx + c Y for any positive integer x, y;

2. If a | B and B | C, A | C;

3. Set M! = 0, then a | B, if and only when Ma | MB;

4. If a | B and B | A, A = + _ B;

5. If a | B and B! = 0, then | A | <=| B |;

3. Prime Number definition:

If a positive integer A is greater than 1 and can only be divisible by 1, A is called a prime number. If a> 1 is not a prime number,

A is the sum of values. Prime numbers are also called prime numbers.

4. Nature of prime number and combination:

1) if D> 1, p is a prime number and d | P, D = P;

2) If P is a prime number, and p | AB, p | A or p | B;

3) The Union number must have a prime factor, that is, if a union a is set, there is a prime number P, making p |;

5. Basic arithmetic Theorem

If a> 1 is set, a = p1 ^ R1 * P2 ^ R2 * P3 ^ R3 * P4 ^ R4 *......;

P1, P2, P3,... PN are different prime numbers, R1, R1, and positive integers ..

The expression in the theorem can be called the prime factor decomposition of:

For example, 24 = 2*2 ** 2*3;

Inference:

D = p1 ^ S1 * P2 ^ S2 * P3 * S3 ^ P4 * s4 ....

0 <= SI <= Ri; I = 1, 2, K ..

This is a necessary and sufficient condition for positive integer d to be a factor.

Number of all factors:

Formula:

N = (1 + R1) * (1 + R2) * (1 + R3) * (1 + R4) *... * (1 + rn );

For example, 99099 = 3 ^ 2*7*11 ^ 2*13

N = 3*2*3*2 = 36;

6. Evaluate the prime number Method

1. Prime Number embedding method.

For example, if you want to calculate a prime number of less than 100, remove the multiple of the prime number in 10 ..

2. The efficiency is very low when we use the definition method.

Code Implementation of the prime number embedding method:

 1 int A[80000], dp[20000];
 2 
 3 const int inf = 65535;
 4 
 5 int prime( )
 6 {
 7     int i, j, k = 0;
 8     A[1] = 1;
 9     for( i = 2; i <= sqrt(inf); i++)
10        for ( j = 2 * i; j <= inf; j += i)
11            A[j] = 1;
12     for( i = 1; i <= inf; i++)
13         if(!A[i])
14             dp[k++] = i;
15     return k;
16 }