Prime Numbers in Discrete Mathematics

Overview :
An integer p>1 is called a prime number, or prime if the only positive divisors of p are 1 and p. An integer q>1 that is not prime is called composite.

Example –
The integers 2,3,5,7 and 11 are prime numbers, and the integers 4,6,8, and 9 are composite.

Theorem-1:
An integer p>1 is prime if and only if for all integers a and b, p divides ab implies either p divides a or p divides b.

Example – 
Consider the integer 12.Now 12 divides 120 = 30 x 4 but 12|30 and 12|4.Hence,12 is not prime.

Theorem-2 :
Every integer n>=2 has a prime factor.

Theorem-3 : 
If n is a composite integer, then n has a prime factor not exceeding √n.

Example-1 :
Determine which of the following integers are prime?

a) 293 b) 9823

Solution –

1. We first find all primes p such that p²< = 293.These primes are 2,3,5,7,11,13 and 17.Now, none of these primes divide 293. Hence, 293 is a prime.
2. We consider primes p such that p2< = 9823.These primes are 2,3,5,7,11,13,17, etc. None of 2,3,5,7 can divide 9823. However,11 divides 9823.Hence, 9823 is not a prime.

Example-2 :
Let n be a positive integer such that n²-1 is prime. Then n =?

Solution – 
We can write, n²-1 = (n-1)(n²+n+1). Because n³-1 is prime, either n-1 = 1 or n²+n+1 = 1.Now n>=1, So n²+n+1 > 1,i.e., n²+n+1 != 1.Thus, we must have n-1 = 1.This implies that n=2.

Example-3 :
Let p be a prime integer such that gcd(a, p³)=p and gcd(b,p⁴)=p. Find gcd(ab,p⁷).

Solution – 
By the given condition, gcd(a,p³)=p. Therefore, p | a. Also, p²|a.(For if p²| a, then gcd (a,p³)>=p²>p, which is a contradiction.) Now a can be written as a product of prime powers. Because p|a and p²| a, it follows that p appears as a factor in the prime factorization of a, but p^k, where k>=2, does not appear in that prime factorization. Similarly ,gcd(b,p⁴)=p implies that p|b and p²|b. As before, it follows that p appears as a factor in the prime factorization of a, but p^k, where k>=2, does not appear in that prime factorization. It now follows that p²|ab and p³|ab. Hence, gcd(b,p⁷) = p² .

Primality Test Algorithm :

for i: [2,N-1]
  if i divides N
     return "Composite"
     
return "Prime"        

Example –
Let’s take an example and makes the algorithm more efficient, 36=

1×36

2×18

3×12

(a=4)x(b=9)

6×6

9×4 (repeated)

12×3 (repeated)

18×2 (repeated)

36×1 (repeated)

Take the inputs of a and b until,    
 a<=b
                                                                
 a . b = N
 a . N/a = N

Modified algorithm :

for i : [2,√n]
   if i divides N
     return "Composite"
     
return "Prime"    

C++

// C++ program to check primality of a number
#include <bits/stdc++.h>
using namespace std;

bool is_prime(int n){
   for(int i = 2; i * i <= n; i++){
     if(n % i == 0){
       return false;
       //if the number is composite or not prime.
     }
   }
  return true;
    // number is prime.
}
int main() {

    int n;
    cin >> n;
    cout << is_prime(n) ? "prime" : "composite";
    return 0;
}

Java

// Java program to check primality of a number

/*package whatever //do not write package name here */

import java.util.*;
class prime{
    public Boolean is_prime(int n){
        for(int i = 2; i * i <= n; i++){
            if(n % i == 0){
                return false;
              //if the number is composite or not prime.
            }
        }
        return true;
      // number is prime.
    }

}

class GFG {
    public static void main (String[] args) {
        Scanner sc = new Scanner(System.in);
        int n = sc.nextInt();
        prime ob = new prime();
        System.out.println(ob.is_prime(n) ? "Prime" : "Composite");
    }
}

Python3

# Python program to check primality of a number
import math


def is_prime(n):
    for i in range(2, int(math.sqrt(n))):
        if(n % i == 0):
            return False
        # if the number is composite or not prime.

    return True
    #  number is prime.


if __name__ == "__main__":

    n = int(input())
    if is_prime(n):
        print("prime")
    else:
        print("composite")

    # This code is contributed by rakeshsahni

Javascript

<script>
// JavaScript program to check primality of a number
function is_prime(n)
{
   for(var i = 2; i * i <= n; i++)
   {
     if(n % i == 0){
       return false;

       // if the number is composite or not prime.
     }
   }
  return true;
    // number is prime.
}
    var n;
    document.write(is_prime(n) ? "prime" : "composite");

// This code is contributed by shivanisinghss2110
</script>

Algorithm to find prime numbers :
In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit.

Algorithm Sieve of Eratosthenes is input: an integer n > 1.
output : all prime numbers from 2 through n.

let A be an array of Boolean values, indexed by integers 2 to n,
initially all set to true.
    
    for i = 2, 3, 4, ..., not exceeding √n do
        if A[i] is true
            for j = i2, i2+i, i2+2i, i2+3i, ..., not exceeding n do
                A[j] := false

    return all i such that A[i] is true.