Uncovering the Secrets of Prime Numbers

The Definition of Prime Numbers

A prime number is a positive integer greater than 1 that has no other factors except 1 and itself. For instance, numbers like 2, 3, 5, 7, 11, and 13 are prime because they cannot be divided evenly by any other number except 1 and themselves. On the other hand, 6 is not prime because it can be divided by 2 and 3 (2 x 3 = 6).

Checking for Prime Numbers

So, how do we determine if a number is prime? Let’s take a closer look. Suppose we want to check if a user-inputted integer is prime or not. First, we need to ensure that the number is greater than 1, since numbers less than or equal to 1 are not considered prime.

def is_prime(num):
    if num <= 1:
        return False
    #... rest of the function...

The Search for Factors

Next, we need to check if the number is exactly divisible by any number from 2 to num – 1. If we find a factor within this range, the number is not prime. However, if we don’t find any factors, the number is prime.

def is_prime(num):
    if num <= 1:
        return False
    for i in range(2, num - 1):
        if num % i == 0:
            return False
    return True

Optimizing the Search

But here’s the thing: we can actually narrow down our search range. Instead of searching from 2 to num – 1, we can use a more efficient range.

• One option is to search from 2 to num / 2.
• An even better approach is to search from 2 to num ** 0.5, based on the fact that a composite number must have a factor less than its square root.

def is_prime(num):
    if num <= 1:
        return False
    for i in range(2, int(num ** 0.5) + 1):
        if num % i == 0:
            return False
    return True

By understanding prime numbers and how to identify them, we can unlock new insights into the world of mathematics. Whether you’re a math enthusiast or just starting to explore the subject, prime numbers are an essential concept to grasp.