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Prime Numbers in Discrete Mathematics

  • Difficulty Level : Basic
  • Last Updated : 27 Aug, 2021

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.

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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 pk, 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 pk, 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.



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