What Is a Prime Number?

Every positive integer greater than 1 falls into one of two categories:

One of the oldest and most efficient ways to find all prime numbers up to a specific limit is the Sieve of Eratosthenes, invented by an ancient Greek mathematician. Here is how it works:

1. Write down a list of numbers starting from 2 up to your limit (e.g., 2 to 30).
2. Circle the first number in the list (2), which is prime. Then, cross out all multiples of 2 (4, 6, 8, etc.) because they are composite.
3. Move to the next uncrossed number (3). Circle it as prime, and cross out all of its remaining multiples (9, 15, 21, etc.).
4. Repeat this process for the next uncrossed numbers (5, 7, etc.).
5. Once you have checked up to the square root of your limit, all remaining uncrossed numbers on your list are prime numbers.

To determine if a larger number is prime without drawing a sieve, you can use the method of trial division:

Divide the target number by every prime number less than or equal to its square root. If any division results in a whole number (no remainder), the number is composite. If none of these primes divide the number evenly, the number is prime.