Check if a number is Primorial Prime or not

Given a positive number N, the task is to check if N is a primorial prime number or not. Print ‘YES’ if N is a primorial prime number otherwise print ‘NO.

Primorial Prime: In Mathematics, A Primorial prime is a prime number of the form pn# + 1 or pn# – 1 , where pn# is the primorial of pn i.e the product of first n prime numbers.

Examples:



Input : N = 5
Output : YES
5 is Primorial prime of the form pn - 1  
for n=2, Primorial is 2*3 = 6
and 6-1 =5.

Input : N = 31
Output : YES
31 is Primorial prime of the form pn + 1  
for n=3, Primorial is 2*3*5 = 30
and 30+1 = 31.

The First few Primorial primes are:

2, 3, 5, 7, 29, 31, 211, 2309, 2311, 30029

Prerequisite:

Approach:

  1. Generate all prime number in the range using Sieve of Eratosthenes.
  2. Check if n is prime or not, If n is not prime Then print No
  3. Else, starting from first prime (i.e 2 ) start multiplying next prime number and keep checking if product + 1 = n or product – 1 = n or not
  4. If either product+1=n or product-1=n, then n is a Primorial Prime Otherwise not.

Below is the implementation of above approach:

C++

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// CPP program to check Primorial Prime
  
#include <bits/stdc++.h>
using namespace std;
  
#define MAX 10000
  
vector<int> arr;
  
bool prime[MAX];
  
// Function to generate prime numbers
void SieveOfEratosthenes()
{
    // Create a boolean array "prime[0..n]" and initialize
    // make all entries of boolean array 'prime'
    // as true. A value in prime[i] will
    // finally be false if i is Not a prime, else true.
  
    memset(prime, true, sizeof(prime));
  
    for (int p = 2; p * p < MAX; p++) {
        // If prime[p] is not changed, then it is a prime
  
        if (prime[p] == true) {
  
            // Update all multiples of p
            for (int i = p * 2; i < MAX; i += p)
                prime[i] = false;
        }
    }
  
    // store all prime numbers
    // to vector 'arr'
    for (int p = 2; p < MAX; p++)
        if (prime[p])
            arr.push_back(p);
}
  
// Function to check the number for Primorial prime
bool isPrimorialPrime(long n)
{
    // If n is not prime Number
    // return flase
    if (!prime[n])
        return false;
  
    long long product = 1;
    int i = 0;
  
    while (product < n) {
  
        // Multiply next prime number
        // and check if product + 1 = n or Product-1 =n
        // holds or not
        product = product * arr[i];
  
        if (product + 1 == n || product - 1 == n)
            return true;
  
        i++;
    }
  
    return false;
}
  
// Driver code
int main()
{
    SieveOfEratosthenes();
  
    long n = 31;
  
    // Check if n is Primorial Prime
    if (isPrimorialPrime(n))
        cout << "YES\n";
    else
        cout << "NO\n";
  
    return 0;
}

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Java

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// Java program to check Primorial prime
  
import java.util.*;
  
class GFG {
  
    static final int MAX = 1000000;
    static Vector<Integer> arr = new Vector<Integer>();
    static boolean[] prime = new boolean[MAX];
  
    // Function to get the prime numbers
    static void SieveOfEratosthenes()
    {
  
        // make all entries of boolean array 'prime'
        // as true. A value in prime[i] will
        // finally be false if i is Not a prime, else true.
  
        for (int i = 0; i < MAX; i++)
            prime[i] = true;
  
        for (int p = 2; p * p < MAX; p++) {
  
            // If prime[p] is not changed, then it is a prime
            if (prime[p] == true) {
  
                // Update all multiples of p
                for (int i = p * 2; i < MAX; i += p)
                    prime[i] = false;
            }
        }
  
        // store all prime numbers
        // to vector 'arr'
        for (int p = 2; p < MAX; p++)
            if (prime[p])
                arr.add(p);
    }
  
    // Function to check the number for Primorial prime
    static boolean isPrimorialPrime(int n)
    {
        // If n is not prime
        // Then return false
        if (!prime[n])
            return false;
  
        long product = 1;
        int i = 0;
        while (product < n) {
  
            // Multiply next prime number
            // and check if product + 1 = n or product -1=n
            // holds or not
            product = product * arr.get(i);
  
            if (product + 1 == n || product - 1 == n)
                return true;
  
            i++;
        }
  
        return false;
    }
  
    // Driver Code
    public static void main(String[] args)
    {
        SieveOfEratosthenes();
  
        int n = 31;
  
        if (isPrimorialPrime(n))
            System.out.println("YES");
        else
            System.out.println("NO");
    }
}

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Python 3

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# Python3 Program to check Primorial Prime 
  
# from math lib import sqrt method
from math import sqrt
  
MAX = 100000
  
# Create a boolean array "prime[0..n]" 
# and initialize make all entries of 
# boolean array 'prime' as true. 
# A value in prime[i] will finally be 
# false if i is Not a prime, else true. 
prime = [True] * MAX
  
arr = []
  
# Utility function to generate
# prime numbers 
def SieveOfEratosthenes() :
  
    for p in range(2, int(sqrt(MAX)) + 1) :
  
        # If prime[p] is not changed, 
        # then it is a prime 
        if prime[p] == True :
  
            # Update all multiples of p 
            for i in range(p * 2 , MAX, p) :
                prime[i] = False
  
    # store all prime numbers 
    # to list 'arr' 
    for p in range(2, MAX) :
  
        if prime[p] :
            arr.append(p)
      
# Function to check the number 
# for Primorial prime 
def isPrimorialPrime(n) :
  
    # If n is not prime Number 
    # return flase 
    if not prime[n] :
        return False
  
    product, i = 1, 0
  
    # Multiply next prime number 
    # and check if product + 1 = n 
    # or Product-1 = n holds or not 
    while product < n :
  
        product *= arr[i]
  
        if product + 1 == n or product - 1 == n :
            return True
  
        i += 1
  
    return False
  
# Driver code
if __name__ == "__main__" :
      
    SieveOfEratosthenes()
      
    n = 31
  
    # Check if n is Primorial Prime 
    if (isPrimorialPrime(n)) :
        print("YES"
    else :
        print("NO"
      
# This code is contributed by ANKITRAI1

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C#

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// c# program to check Primorial prime 
using System;
using System.Collections.Generic;
  
public class GFG
{
  
    public const int MAX = 1000000;
    public static List<int> arr = new List<int>();
    public static bool[] prime = new bool[MAX];
  
    // Function to get the prime numbers 
    public static void SieveOfEratosthenes()
    {
  
        // make all entries of boolean array 'prime' 
        // as true. A value in prime[i] will 
        // finally be false if i is Not a prime, else true. 
  
        for (int i = 0; i < MAX; i++)
        {
            prime[i] = true;
        }
  
        for (int p = 2; p * p < MAX; p++)
        {
  
            // If prime[p] is not changed, then it is a prime 
            if (prime[p] == true)
            {
  
                // Update all multiples of p 
                for (int i = p * 2; i < MAX; i += p)
                {
                    prime[i] = false;
                }
            }
        }
  
        // store all prime numbers 
        // to vector 'arr' 
        for (int p = 2; p < MAX; p++)
        {
            if (prime[p])
            {
                arr.Add(p);
            }
        }
    }
  
    // Function to check the number for Primorial prime 
    public static bool isPrimorialPrime(int n)
    {
        // If n is not prime 
        // Then return false 
        if (!prime[n])
        {
            return false;
        }
  
        long product = 1;
        int i = 0;
        while (product < n)
        {
  
            // Multiply next prime number 
            // and check if product + 1 = n or product -1=n 
            // holds or not 
            product = product * arr[i];
  
            if (product + 1 == n || product - 1 == n)
            {
                return true;
            }
  
            i++;
        }
  
        return false;
    }
  
    // Driver Code 
    public static void Main(string[] args)
    {
        SieveOfEratosthenes();
  
        int n = 31;
  
        if (isPrimorialPrime(n))
        {
            Console.WriteLine("YES");
        }
        else
        {
            Console.WriteLine("NO");
        }
    }
}
  
// This code is contributed by Shrikant13

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PHP

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<?php
// PHP Program to check Primorial Prime 
$MAX = 100000;
  
// Create a boolean array "prime[0..n]" 
// and initialize make all entries of 
// boolean array 'prime' as true. 
// A value in prime[i] will finally be 
// false if i is Not a prime, else true. 
$prime = array_fill(0, $MAX, true);
  
$arr = array();
  
// Utility function to generate
// prime numbers 
function SieveOfEratosthenes()
{
    global $MAX, $prime, $arr;
    for($p = 2; $p <= (int)(sqrt($MAX)); $p++)
    {
  
        // If prime[p] is not changed, 
        // then it is a prime 
        if ($prime[$p] == true)
  
            // Update all multiples of p 
            for ($i = $p * 2; $i < $MAX; $i += $p)
                $prime[$i] = false;
    }
  
    // store all prime numbers 
    // to list 'arr' 
    for ($p = 2; $p < $MAX; $p++)
        if ($prime[$p])
            array_push($arr, $p);
}
      
// Function to check the number 
// for Primorial prime 
function isPrimorialPrime($n)
{
    global $MAX, $prime, $arr;
      
    // If n is not prime Number 
    // return flase 
    if(!$prime[$n])
        return false;
  
    $product = 1;
    $i = 0;
  
    // Multiply next prime number 
    // and check if product + 1 = n 
    // or Product-1 = n holds or not 
    while ($product < $n)
    {
        $product *= $arr[$i];
  
        if ($product + 1 == $n || 
            $product - 1 == $n )
            return true;
  
        $i += 1;
    }
  
    return false;
}
  
// Driver code
SieveOfEratosthenes();
  
$n = 31;
  
// Check if n is Primorial Prime 
if (isPrimorialPrime($n))
    print("YES");
else
    print("NO"); 
  
// This code is contributed by mits

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Output:

YES


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