How Can a PHP Function Efficiently Determine if a Number is Prime?

A Formula to Find Prime Numbers in a Loop

This question seeks to identify prime numbers using a looping mechanism. Specifically, the question aims to create a PHP function to find prime numbers efficiently.

To understand the process, let's introduce the concept of prime numbers. Prime numbers are whole numbers greater than 1 that are not divisible by any other whole number except 1 and themselves.

This definition suggests a straightforward way to check for primality: divide the number by all integers from 2 to the square root of the number. If any of these divisions have a remainder, the number is prime.

The PHP function provided in the question's answer adheres to this concept:

<code class="php">function isPrime($num) {
    //1 is not prime. See: http://en.wikipedia.org/wiki/Prime_number#Primality_of_one
    if($num == 1)
        return false;

    //2 is prime (the only even number that is prime)
    if($num == 2)
        return true;

    /**
     * if the number is divisible by two, then it's not prime and it's no longer
     * needed to check other even numbers
     */
    if($num % 2 == 0) {
        return false;
    }

    /**
     * Checks the odd numbers. If any of them is a factor, then it returns false.
     * The sqrt can be an aproximation, hence just for the sake of
     * security, one rounds it to the next highest integer value.
     */
    $ceil = ceil(sqrt($num));
    for($i = 3; $i <= $ceil; $i = $i + 2) {
        if($num % $i == 0)
            return false;
    }

    return true;
}</code>

This function utilizes an array to store the factors of the number and checks the remainders of the divisions. If any remainder is zero, it indicates the presence of a factor, rendering the number non-prime. However, if no factors are found, the number is considered prime.

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