Open In App

Check if a number is Primorial Prime or not

Last Updated : 25 Aug, 2022
Comments
Improve
Suggest changes
Like Article
Like
Report

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++ 
// 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 false
    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;
}
Java 
// 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");
    }
}
Python 3 
# 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 false 
    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
C# 
// 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
PHP 
<?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 false 
    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
JavaScript 
<script>

// Javascript program to check Primorial Prime

var MAX = 10000;

var arr = [];

var prime = Array(MAX).fill(true);

// Function to generate prime numbers
function 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.

    for (var 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 (var i = p * 2; i < MAX; i += p)
                prime[i] = false;
        }
    }

    // store all prime numbers
    // to vector 'arr'
    for (var p = 2; p < MAX; p++)
        if (prime[p])
            arr.push(p);
}

// Function to check the number for Primorial prime
function isPrimorialPrime(n)
{
    // If n is not prime Number
    // return false
    if (!prime[n])
        return false;

    var product = 1;
    var 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
SieveOfEratosthenes();
var n = 31;
// Check if n is Primorial Prime
if (isPrimorialPrime(n))
    document.write( "YES");
else
    document.write("NO");

</script>

Output: 
YES

 

Time Complexity: O(n + MAX3/2)

Auxiliary Space: O(MAX)


Next Article
Sum of all Primes in a given range using Sieve of Eratosthenes

I
ihritik
Improve
Article Tags :
Practice Tags :

Similar Reads

We use cookies to ensure you have the best browsing experience on our website. By using our site, you acknowledge that you have read and understood our Cookie Policy & Privacy Policy
Lightbox