Primality Test

Primality test is a test to determine whether a number is prime or not. There are many different primality tests.

What is a prime number?

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A number is prime if it has exactly two positive whole number divisors, one and itself. On the other hand, a number is composite if it has more than two positive whole number divisors. Some interesting facts about prime numbers are: one is not a prime number since it does not satisfy the definition of being a prime, and there exist no even primes greater than 2.

Now, let’s look at the naive approach

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To find if a number n is prime we could simply check if it divides any numbers below it.

C code:

#include<stdio.h>
#include<stdbool.h>
bool isPrime(long n);

int main(void){
    long n;
    printf("Enter the number for primality test: ");
    scanf("%ld", &n);
    printf("Result: %s\n", isPrime(n)?"Is a Prime.":"Not a Prime.");
}
bool isPrime(long n){
    long i;
    for(i=2; i<n; i++){
        if(n%i == 0){
            printf("It's divisible by %ld.\n", i);
            return false;
        }
    }
    return true;
}

Time complexity: \(O(n)\)

The better approach

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• We only need to check divisibility for values of i that are less or equal to the square root of n 
• Since there exist no even primes greater than 2. Therefore, after checking n is not an even number we can safely increment the value of i by 2 to avoid dividing n by odd numbers.

C code

#include<stdio.h>
#include<stdbool.h>
#include<math.h>
bool isPrime(long n);

int main(void){
    long n;
    printf("Enter the number for primality test: ");
    scanf("%ld", &n);
    printf("Result: %s\n", isPrime(n)?"Is a Prime.":"Not a Prime.");
}
bool isPrime(long n){
    long i;
    if(n<=1) return false;
    if(n==2) return true;
    if(n%2 == 0) return false; //even check

    for(i=3; i<=sqrt(n); i+=2){ //increment by 2 to avoid checking odd nos.
        if(n%i == 0){
            return false;
        }
    }
    return true;
}

Time complexity: \(O(√n)\)