检查数字是否为素数

给定正整数N ，任务是检查给定数字是否为质数。如果给定的数字很好，则打印’ YES ‘，否则打印’ 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, ….

例子：

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 

方法：

1.获取数字N。

2.初始化prev_prime = N-1和next_prime = N + 1

3.在prev_prime大于或等于2时迭代循环。并检查next_prime和prev_prime是否都是不使用素数的素数。

4.如果两者都不是素数，请重复步骤2和3。

5.如果next_prime和prev_prime都是素数，则检查N ^ 2> next_prime。 prev_prime与否。

• 如果不是，则数字不是很好的素数，然后停止执行并返回NO。
• 如果是，则重复步骤2、3、4和5。

以下是上述方法的实现：

C++

// C++ program to check if a number
// is good prime or not
#include
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

 
// 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
 

输出：

YES

时间复杂度： O(n ^3/2 )

辅助空间： O(1)