给定正数N，任务是检查N是否为原始质数。如果N是原始质数，则打印“ YES”，否则打印“ NO”。
素数素数：在数学中，素数素数是形式为p n ＃+ 1或p n ＃– 1的素数，其中p n ＃是p n的素数，即前n个素数的乘积。
例子：
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.
前几个主要素数是：
2, 3, 5, 7, 29, 31, 211, 2309, 2311, 30029
先决条件：
- 原始的
- Eratosthenes筛
方法：
- 使用Eratosthenes筛网生成该范围内的所有素数。
- 检查n是否为质数，如果n不是质数，则打印否
- 否则，从第一个质数(即2)开始乘以下一个质数，并继续检查乘积+ 1 = n或乘积– 1 = n
- 如果product + 1 = n或product-1 = n ，则n是基本素数，否则不是。
下面是上述方法的实现：
C++
// CPP program to check Primorial Prime
#include
using namespace std;
#define MAX 10000
vector 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;
} Java
// Java program to check Primorial prime
import java.util.*;
class GFG {
static final int MAX = 1000000;
static Vector arr = new Vector();
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 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 ANKITRAI1C#
// c# program to check Primorial prime
using System;
using System.Collections.Generic;
public class GFG
{
public const int MAX = 1000000;
public static List arr = new List();
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
输出：
YES