素数判断

School Method
A simple solution is to iterate through all numbers from 2 to n-1 and for every number check if it divides n. If we find any number that divides, we return false.

Below is the implementation of this method.

// 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; 
} 

Optimized School Method
We can do following optimizations:

1. Instead of checking till n, we can check till √n because a larger factor of n must be a multiple of smaller factor that has been already checked.
2. The algorithm can be improved further by observing that all primes are of the form 6k ± 1, with the exception of 2 and 3. This is because all integers can be expressed as (6k + i) for some integer k and for i = -1, 0, 1, 2, 3, or 4; 2 divides (6k + 0), (6k + 2), (6k + 4); and 3 divides (6k + 3). So a more efficient method is to test if n is divisible by 2 or 3, then to check through all the numbers of form 6k ± 1. (Source: wikipedia)

   Below is the implementation of this solution.

// 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; 
  
    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(11)?  cout << " true\n": cout << " false\n"; 
    isPrime(15)?  cout << " true\n": cout << " false\n"; 
    return 0; 
}