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Check whether a number is Good prime or not
  • Difficulty Level : Medium
  • Last Updated : 08 Mar, 2021
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Given a positive integer N, the task is to check whether the given number is good prime or not. If the given number is good prime print ‘YES’ Otherwise Print ‘NO’. 

Good Prime: In Mathematics, a good prime is a prime number whose square is greater than the product of any two primes at the same number of positions before and after it in the sequence of primes. In other word, A prime Pn is said to be good prime if it Pn^2 > P(n-i) . P(n+1)     for every 1 <= i < n.

The first few good primes are: 5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, ….

Examples:

Input: N = 5 
Output: YES 
Explanation: 5 is a good prime number 
since 5^2 = 25 is greater than 3.7 = 21 
and 2.11 = 22. 
 



Input:  N = 20 
Output: NO 

 

Approach:

1. Get the number N.

2. Initialise prev_prime = N-1 and next_prime = N+1

3. Iterate the loop while prev_prime is greater than or equal to 2. And check for both next_prime and prev_prime are prime of not using prime number.

4. If both are not prime, then repeat step 2 and 3.

5. If both next_prime and prev_prime are prime, then check N^2 > next_prime . prev_prime or not.

  • If Not then number is not good prime and stop the execution and return NO.
  • If Yes then repeat the step 2, 3, 4 and 5.

Below is the implementaion of the above approach:

C++




// C++ program to check if a number
// is good prime or not
#include<bits/stdc++.h>
using namespace std;
 
// Function to check if a
// number is Prime or not
bool isPrime (int n)
{
 
    // Corner cases
    if (n <= 1)
        return false;
    if (n <= 3)
        return true;
 
    // This is checked so that we can
    // skip middle five numbers in loop
    if (n % 2 == 0 || n % 3 == 0)
        return false;
 
    for(int i = 5; i * i <= n; i += 6)
    {
       if (n % i == 0 || n % (i + 2) == 0)
           return false;
    }
    return true;
}
 
// Function to check if the
// given number is Good prime
bool isGoodprime (int n)
{
 
    // Smallest good prime is 5
    // So the number less than 5
    // can not be a Good prime
 
    if (n < 5)
        return false;
 
    int prev_prime = n - 1;
    int next_prime = n + 1;
 
    while (prev_prime >= 2)
    {
         
        // Calculate first prime number < n
        while (!isPrime(prev_prime))
        {
            prev_prime--;
        }
 
        // Calculate first prime number > n
        while (!isPrime(next_prime))
        {
            next_prime++;
        }
 
        // Check if product of next_prime
        // and prev_prime is less than n^2
        if ((prev_prime * next_prime) >= n * n)
            return false;
 
        prev_prime -= 1;
        next_prime += 1;
    }
    return true;
}
 
// Driver code
int main()
{
    int n = 11;
 
    if (isGoodprime(n))
        cout << "YES";
    else
        cout << "NO";
 
    return 0;
}
 
// This code is contributed by himanshu77

Java




// Java program to check if a number is
// good prime or not
class GFG{
 
// Function to check if a
// number is prime or not
static boolean isPrime(int n)
{
     
    // Corner cases
    if (n <= 1)
        return false;
    if (n <= 3)
        return true;
 
    // This is checked so that we can skip
    // middle five numbers in below loop
    if (n % 2 == 0 || n % 3 == 0)
        return false;
 
    for(int i = 5; i * i <= n; i = i + 6)
    {
       if (n % i == 0 || n % (i + 2) == 0)
       {
           return false;
       }
    }
    return true;
}
 
// Function to check if the given
// number is good prime or not
static boolean isGoodrprime(int n)
{
 
    // Smallest good prime is 5
    // So the number less than 5
    // can not be a good prime
 
    if (n < 5)
        return false;
 
    int prev_prime = n - 1;
    int next_prime = n + 1;
 
    while (prev_prime >= 2)
    {
         
        // Calculate first prime number < n
        while (!isPrime(prev_prime))
        {
            prev_prime--;
        }
 
        // Calculate first prime number > n
        while (!isPrime(next_prime))
        {
            next_prime++;
        }
 
        // Check if product of next_prime
        // and prev_prime
        // is less than n^2
        if ((prev_prime * next_prime) >= n * n)
            return false;
 
        prev_prime -= 1;
        next_prime += 1;
    }
    return true;
}
 
// Driver code
public static void main(String []args)
{
    int n = 11;
     
    if (isGoodrprime(n))
        System.out.println("YES");
    else
        System.out.println("NO");
}
}
 
// This code is contributed by amal kumar choubey

Python3




# Python3 program to check if a number is
# good prime or not
 
# Utility function to check
# if a number is prime or not
def isPrime(n):
     
    # Corner cases
    if (n <= 1):
        return False
    if (n <= 3):
        return True
 
    # This is checked so that we can skip
    # middle five numbers in below loop
    if (n % 2 == 0 or n % 3 == 0):
        return False
 
    i = 5
    while (i * i <= n):
        if (n % i == 0 or n % (i + 2) == 0):
            return False
        i = i + 6
    return True
 
# Function to check if the given number
# is good prime or not
def isGoodrPrime(n):
 
    # Declaring variables as global
    global next_prime, prev_prime
 
    # Smallest good prime is 5
    # So the number less than 5
    # can not be a good prime
    if(n < 5):
        return False
 
    # Initialize previous_prime to n - 1
    # and next_prime to n + 1
    prev_prime = n - 1
    next_prime = n + 1
 
    while(prev_prime >= 2):
 
        # Calculate first prime number < n
        while (not isPrime(prev_prime)):
            prev_prime -= 1
 
        # Calculate first prime number > n
        while(not isPrime(next_prime)):
            next_prime += 1
 
        # Check if product of next_prime
        # and prev_prime
        # is less than n^2
        if((prev_prime * next_prime) >= n * n):
            return False
 
        prev_prime -= 1
        next_prime += 1
 
    return True
 
# Driver code
if __name__ == '__main__':
 
    n = 11
 
    if(isGoodrPrime(n)):
        print("Yes")
    else:
        print("No")
 
# This code is contributed by Shivam Singh

C#




// C# program to check if a number is
// good prime  or not
 
using System;
class GFG {
 
    // Function to check if a
    // number is prime or not
    static bool isPrime(int n)
    {
        // Corner cases
        if (n <= 1)
            return false;
        if (n <= 3)
            return true;
 
        // This is checked so that we can skip
        // middle five numbers in below loop
        if (n % 2 == 0 || n % 3 == 0)
            return false;
 
        for (int i = 5; i * i <= n; i = i + 6) {
            if (n % i == 0 || n % (i + 2) == 0) {
                return false;
            }
        }
        return true;
    }
 
    // Function to check
    // if the given number is good prime or not
    static bool isGoodrprime(int n)
    {
 
        // Smallest good prime is 5
        // So the number less than 5
        // can not be a good prime
 
        if (n < 5)
            return false;
 
        int prev_prime = n - 1;
        int next_prime = n + 1;
 
        while (prev_prime >= 2) {
            // Calculate first prime number < n
            while (!isPrime(prev_prime)) {
                prev_prime--;
            }
 
            // Calculate first prime number > n
            while (!isPrime(next_prime)) {
                next_prime++;
            }
 
            // check if product of next_prime
            // and prev_prime
            // is less than n^2
 
            if ((prev_prime * next_prime)
                >= n * n)
                return false;
 
            prev_prime -= 1;
            next_prime += 1;
        }
        return true;
    }
 
    public static void Main()
    {
 
        int n = 11;
        if (isGoodrprime(n))
            Console.WriteLine("YES");
        else
            Console.WriteLine("NO");
    }
}

Javascript




<script>
 
// Javascript program to check if a number
// is good prime or not
 
// Function to check if a
// number is Prime or not
function isPrime (n)
{
 
    // Corner cases
    if (n <= 1)
        return false;
    if (n <= 3)
        return true;
 
    // This is checked so that we can
    // skip middle five numbers in loop
    if (n % 2 == 0 || n % 3 == 0)
        return false;
 
    for(let i = 5; i * i <= n; i += 6)
    {
    if (n % i == 0 || n % (i + 2) == 0)
        return false;
    }
    return true;
}
 
// Function to check if the
// given number is Good prime
 
function isGoodprime (n)
{
 
    // Smallest good prime is 5
    // So the number less than 5
    // can not be a Good prime
 
    if (n < 5)
        return false;
 
    let prev_prime = n - 1;
    let next_prime = n + 1;
 
    while (prev_prime >= 2)
    {
         
        // Calculate first prime number < n
        while (!isPrime(prev_prime))
        {
            prev_prime--;
        }
 
        // Calculate first prime number > n
        while (!isPrime(next_prime))
        {
            next_prime++;
        }
 
        // Check if product of next_prime
        // and prev_prime is less than n^2
        if ((prev_prime * next_prime) >= n * n)
            return false;
 
        prev_prime -= 1;
        next_prime += 1;
    }
    return true;
}
 
// Driver code
 
    let n = 11;
 
    if (isGoodprime(n))
        document.write("YES");
    else
        document.write("NO");
 
 
// This code is contributed by Mayank Tyagi
 
</script>
Output: 
YES

 

Time Complexity: O(n3/2)

Auxiliary Space: O(1)

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