C++ Program to check Prime Number

Given a positive integer, check if the number is prime or not. A prime is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Examples:

Input:  n = 11
Output: true

Input:  n = 15
Output: false

Input:  n = 1
Output: false

Naive approach:

CPP

// A school method based C++ program to check if a
// number is prime
#include <bits/stdc++.h>
using namespace std;

bool isPrime(int n)
{
    // Corner case
    if (n <= 1)
        return false;

    // Check from 2 to n-1
    for (int i = 2; i < n; i++)
        if (n % i == 0)
            return false;

    return true;
}

// Driver Program to test above function
int main()
{
    isPrime(11) ? cout << " true\n" : cout << " false\n";
    isPrime(15) ? cout << " true\n" : cout << " false\n";
    return 0;
}

Output

 true
 false

Time Complexity: O(n) 
Auxiliary Space: O(1)

Optimized School Method:

CPP

// A optimized school method based C++ program to check
// if a number is prime
#include <bits/stdc++.h>
using namespace std;

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;

    // Using concept of prime number
    // can be represented in form of
    // (6*n + 1) or(6*n - 1) hence
    // we have to go for every multiple of 6 and
    // prime number would always be 1 less or 1 more than
    // the multiple of 6.
    for (int i = 5; i * i <= n; i = i + 6)
        if (n % i == 0 || n % (i + 2) == 0)
            return false;

    return true;
}

// Driver Program to test above function
int main()
{
    isPrime(4191) ? cout << " true\n" : cout << " false\n";
    isPrime(15) ? cout << " true\n" : cout << " false\n";
    return 0;
}

Output

 false
 false

Time Complexity: O(sqrt(n)) 
Auxiliary Space: O(1)

Please refer complete article on Primality Test | Set 1 (Introduction and School Method) for more details!

Using Wilson’s theorem: 

Given a number N, the task is to check if it is prime or not using Wilson Primality Test. Print ‘1’ if the number is prime, else print ‘0’.
Wilson’s theorem states that a natural number p > 1 is a prime number if and only if

    (p - 1) ! ≡  -1   mod p 
OR  (p - 1) ! ≡  (p-1) mod p

Example : 

Input: p = 15
Output: Yes
(p-1)! = 14! = 87178291200 
87178291200 % 15  = 0

C++

// C++ implementation to check if a number is
// prime or not using Wilson Primality Test
#include <bits/stdc++.h>
using namespace std;

// Function to calculate the factorial
long long fact(const int& p)
{
    if (p <= 1)
        return 1;
    return p * fact(p - 1);
}

// Function to check if the
// number is prime or not
bool isPrime(const long long& p)
{
    if (p == 4)
        return false;

    //  (p - 1) ! ≡  (p-1) mod p
    long long a = fact(p - 1) % p;
    if (a == p - 1)
        return true;
    return false;
}

// Driver code
int main()
{
    if (isPrime(13) == true)
        cout << "True" << endl;
    else
        cout << "false" << endl;

    if (isPrime(15) == true)
        cout << "True" << endl;
    else
        cout << "false" << endl;
    return 0;
}
// this code is contributed by devendra solunke

Output

True
false

Time Complexity : O(n) 
Auxiliary Space :  O(n)

Please refer complete article on Implementation of Wilson Primality test for more details!