Check if a number is Prime, Semi-Prime or Composite for very large numbers

Given a very large number N (> 150), the task is to check whether this number is Prime, Semi-Prime or Composite.

Example:

Input: N = 90000000
Output: Not Prime
Explanation:
we have (N-1)%6 = 89999999%6 = 1 and
(N+1)%6 = 90000001%6 = 5
Since n-1 and n+1 is not divisible by 6
Therefore N = 90000000 is Not Prime

Input: N = 7894561
Output: Semi-Prime
Explanation:
Here N = 7894561 = 71*111191
Since 71 & 111191 are prime, therefore 7894561 is Semi Prime

Approach:



  • It can be observed that if n is a Prime Number then n+1 or n-1 will be divisible by 6
  • If a number n exists such that neither n+1 nor n-1 is divisble by 6 then n is not a prime number
  • If a number n exists such that either n+1 or n-1 is divisible by 6 then n is either a prime or a semiprime number
  • To differentiate between prime and semi-prime, the following method is used:
    • If N is semi prime then,
      N = p*q  ....................(1)
      where p & q are primes.
      
    • Then from Goldbach Conjecture:
      p + q must be even
      i.e, p + q = 2*n for any positive integer n
      
    • Therefore solving for p & q will give
      p = n - sqrt(n2 - N)
      q = n + sqrt(n2 - N)
      
    • Let n2 – N be perfect square, Then
      n2 - N = m2, .................(2)
      for any positive integer m 
      
    • Solving Equations (1) & (2) we get
      m = (q-p)/2
      n = (p+q)/2
      
    • Now if equation (1) & (2) meets at some point, then there exists a pair (p, q) such that the number N is semiprime otherwise N is prime.
  • Equation(2) forms Pythagorean Triplet

  • The solution expected varies on the graph

Pseudo code:

  • Input a number N and if N – 1 and N + 1 is not divisible by 6 then the number N is Not Prime. else it is prime or semi-prime
  • If n-1 or n+1 is divisible by 6 then iterate in the range(sqrt(N) + 1, N) and find a pair (p, q) such that p*q = N by below formula:
    p = i - sqrt(i*i - N)
    q = n/p
    where i = index in range(sqrt(N) + 1, N)
  • If p*q = N then the number N is semi prime, else it is prime

Below is the implementation of the above approach:

Java

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import static java.lang.Math.sqrt;
  
public class Primmefunc {
  
    public static void prime(long n)
    {
        int flag = 0;
  
        // checking divisibilty by 6
        if ((n + 1) % 6 != 0 && (n - 1) % 6 != 0) {
            System.out.println("Not Prime");
        }
        else {
  
            // breakout if number is perfect square
            double s = sqrt(n);
            if ((s * s) == n) {
                System.out.println("Semi-Prime");
            }
            else {
                long f = (long)s;
                long l = (long)((f * f));
  
                // Iterating over to get the
                // closest average value
                for (long i = f + 1; i < l; i++) {
  
                    // 1st Factor
                    long p = i - (long)(sqrt((i * i) - (n)));
  
                    // 2nd Factor
                    long q = n / p;
  
                    // To avoid Convergence
                    if (p < 2 || q < 2) {
                        break;
                    }
  
                    // checking semi-prime condition
                    if ((p * q) == n) {
                        flag = 1;
                        break;
                    }
  
                    // If convergence found
                    // then number is semi-prime
                    else {
  
                        // convergence not found
                        // then number is prime
                        flag = 2;
                    }
                }
  
                if (flag == 1) {
                    System.out.println("Semi-Prime");
                }
                else if (flag == 2) {
  
                    System.out.println("Prime");
                }
            }
        }
    }
  
    public static void main(String[] args)
    {
        // Driver code
        // Entered number should be greater
        // than 300 to avoid Convergence of
        // second factor to 1
        prime(8179);
        prime(7894561);
        prime(90000000);
        prime(841);
        prime(22553);
        prime(1187);
    }
//written by Rushil Jhaveri
}

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CPP

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#include<bits/stdc++.h>
using namespace std ;
  
void prime(long n)
{
    int flag = 0;
  
    // checking divisibilty by 6
    if ((n + 1) % 6 != 0 && (n - 1) % 6 != 0) 
    {
        cout << ("Not Prime") << endl;
    }
    else
    {
  
        // breakout if number is perfect square
        double s = sqrt(n);
        if ((s * s) == n) 
        {
            cout<<("Semi-Prime")<<endl;
        }
        else 
        {
            long f = (long)s;
            long l = (long)((f * f));
  
            // Iterating over to get the
            // closest average value
            for (long i = f + 1; i < l; i++)
            {
  
                // 1st Factor
                long p = i - (long)(sqrt((i * i) - (n)));
  
                // 2nd Factor
                long q = n / p;
  
                // To avoid Convergence
                if (p < 2 || q < 2) 
                {
                    break;
                }
  
                // checking semi-prime condition
                if ((p * q) == n)
                {
                    flag = 1;
                    break;
                }
  
                // If convergence found
                // then number is semi-prime
                else 
                {
  
                    // convergence not found
                    // then number is prime
                    flag = 2;
                }
            }
  
            if (flag == 1) 
            {
                cout<<("Semi-Prime")<<endl;
            }
            else if (flag == 2)
            {
  
                cout<<("Prime")<<endl;
            }
        }
    }
}
  
// Driver code
int main()
{
      
    // Entered number should be greater
    // than 300 to avoid Convergence of
    // second factor to 1
    prime(8179);
    prime(7894561);
    prime(90000000);
    prime(841);
    prime(22553);
    prime(1187);
}
  
// This code is contributed by Rajput-Ji

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Python3

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def prime(n):
    flag = 0;
  
    # checking divisibilty by 6
    if ((n + 1) % 6 != 0 and (n - 1) % 6 != 0):
        print("Not Prime");
    else:
  
        # breakout if number is perfect square
        s = pow(n, 1/2);
        if ((s * s) == n):
            print("Semi-Prime");
        else:
            f = int(s);
            l = int(f * f);
  
            # Iterating over to get the
            # closest average value
            for i in range(f + 1, l):
  
                # 1st Factor
                p = i - (pow(((i * i) - (n)), 1/2));
  
                # 2nd Factor
                q = n // p;
  
                # To avoid Convergence
                if (p < 2 or q < 2):
                    break;
                  
                # checking semi-prime condition
                if ((p * q) == n):
                    flag = 1;
                    break;
                  
                # If convergence found
                # then number is semi-prime
                else:
  
                    # convergence not found
                    # then number is prime
                    flag = 2;
                  
            if (flag == 1):
                print("Semi-Prime");
            elif(flag == 2):
  
                print("Prime");
              
# Driver code
if __name__ == '__main__':
  
    # Entered number should be greater
    # than 300 to avoid Convergence of
    # second factor to 1
    prime(8179);
    prime(7894561);
    prime(90000000);
    prime(841);
    prime(22553);
    prime(1187);
  
# This code is contributed by 29AjayKumar

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C#

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using System;
public class Primmefunc 
{
  
    public static void prime(long n)
    {
        int flag = 0;
  
        // checking divisibilty by 6
        if ((n + 1) % 6 != 0 && (n - 1) % 6 != 0)
        {
            Console.WriteLine("Not Prime");
        }
        else 
        {
  
            // breakout if number is perfect square
            double s = Math.Sqrt(n);
            if ((s * s) == n)
            {
                Console.WriteLine("Semi-Prime");
            }
            else 
            {
                long f = (long)s;
                long l = (long)((f * f));
  
                // Iterating over to get the
                // closest average value
                for (long i = f + 1; i < l; i++) 
                {
  
                    // 1st Factor
                    long p = i - (long)(Math.Sqrt((i * i) - (n)));
  
                    // 2nd Factor
                    long q = n / p;
  
                    // To avoid Convergence
                    if (p < 2 || q < 2) 
                    {
                        break;
                    }
  
                    // checking semi-prime condition
                    if ((p * q) == n)
                    {
                        flag = 1;
                        break;
                    }
  
                    // If convergence found
                    // then number is semi-prime
                    else
                    {
  
                        // convergence not found
                        // then number is prime
                        flag = 2;
                    }
                }
  
                if (flag == 1) 
                {
                    Console.WriteLine("Semi-Prime");
                }
                else if (flag == 2)
                {
                    Console.WriteLine("Prime");
                }
            }
        }
    }
  
    // Driver code
    public static void Main(String[] args)
    {
        // Entered number should be greater
        // than 300 to avoid Convergence of
        // second factor to 1
        prime(8179);
        prime(7894561);
        prime(90000000);
        prime(841);
        prime(22553);
        prime(1187);
    }
}
  
// This code is contributed by 29AjayKumar

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Output:

Prime
Semi-Prime
Not Prime
Semi-Prime
Semi-Prime
Prime

Time Complexity: O(N)

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