What Are Prime Numbers?
Numbers that refuse to be divided — and why mathematicians have been obsessed with them for 2,000 years.
The Definition
A prime number is a whole number greater than 1 that can only be divided evenly by 1 and itself. That means no other number goes into it without leaving a remainder. The number 7 is prime because the only way to divide it evenly is 7 ÷ 1 = 7 or 7 ÷ 7 = 1. No other number works.
A number that isn't prime (meaning it CAN be divided by other numbers) is called a composite number. The number 12 is composite because it can be divided by 1, 2, 3, 4, 6, and 12.
Note: the number 1 is neither prime nor composite — it's in its own special category.
The First 25 Prime Numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
Notice anything? The number 2 is the only even prime number. Every other even number can be divided by 2, so they're all composite. After 2, all primes are odd — but not all odd numbers are prime (9, 15, 21, 25 are all odd but not prime).
How to Find Prime Numbers: The Sieve of Eratosthenes
About 2,200 years ago, a Greek mathematician named Eratosthenes invented a brilliant method for finding all prime numbers up to any limit. Here's how it works:
- Write down all numbers from 2 to your limit (say, 100)
- Circle 2 (the first prime) and cross out every multiple of 2 (4, 6, 8, 10, …)
- Circle 3 (the next uncrossed number) and cross out every multiple of 3 (6, 9, 12, 15, …)
- Circle 5 and cross out every multiple of 5
- Continue this process — every number that hasn't been crossed out is prime
It's called a "sieve" because you're filtering out composite numbers, and the primes are what remain — like using a strainer to catch gold nuggets.
Why Primes Matter
Prime numbers are the building blocks of all numbers. Every whole number greater than 1 can be written as a product of prime numbers in exactly one way. This is called the Fundamental Theorem of Arithmetic. For example: 12 = 2 × 2 × 3. The number 60 = 2 × 2 × 3 × 5. This "prime factorization" is unique to each number, like a fingerprint.
Primes also power modern internet security. When you buy something online, your credit card number is protected by encryption algorithms that rely on the fact that multiplying two large prime numbers is easy, but figuring out which two primes were multiplied together is extraordinarily difficult — even for the world's fastest computers.
The largest known prime number (as of early 2025) has over 41 million digits. It would take thousands of pages just to print it. Finding enormous primes is a hobby for some mathematicians and computer scientists — there's even a global volunteer project (GIMPS) where regular people donate their computer's processing power to search for the next record-breaking prime.
Last reviewed: April 2026