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
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 implementation 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 notbool 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 primebool 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 codeint 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 notclass GFG{// Function to check if a// number is prime or notstatic 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 notstatic 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 codepublic 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 notdef 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 codeif __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 notusing 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 notfunction 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 primefunction 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(n)
Auxiliary Space: O(1)
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