Prime Finder

The Sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to a given limit. It works by:

1. Creating a list of all numbers from 2 to the limit
2. Starting with 2, marking all its multiples as composite
3. Moving to the next unmarked number and repeating
4. Continuing until you've processed all numbers up to √limit

All remaining unmarked numbers are prime!

The tool uses trial division: testing if the number is divisible by any integer from 2 to √n.

Why √n? If n = a × b where both a, b > √n, then a × b > n, which is impossible. So at least one factor must be ≤ √n.

Prime factorization breaks a number down into its prime factors. Every integer > 1 can be uniquely expressed as a product of primes.

Example: 60 = 2² × 3 × 5

This is called the Fundamental Theorem of Arithmetic.

By definition, a prime number must have exactly two distinct positive divisors: 1 and itself.

The number 1 only has one divisor (itself), so it doesn't meet this criterion. This definition makes many theorems and formulas work correctly.