Check whether any number is prime, find all primes up to a limit, or get the prime factorization of any number — all with our free prime number calculator. Instant results with explanations for students, teachers, and math enthusiasts.
Is It a Prime Number?
Prime Factorization
List Primes Up To…
What Is a Prime Number?
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23… The number 1 is not prime by definition. 2 is the only even prime number.
The First 20 Prime Numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71
Frequently Asked Questions
Is 1 a prime number?
No. By mathematical convention, 1 is not considered prime. The definition requires a prime to have exactly two distinct factors (1 and itself). The number 1 has only one factor (itself), so it doesn’t qualify. This exclusion keeps prime factorization unique.
Is 2 a prime number?
Yes. 2 is the only even prime number. All other even numbers are divisible by 2, making them composite. 2 is also the smallest prime number.
How do you check if a large number is prime?
Test divisibility by all primes up to the square root of the number. If no prime divides it evenly, it’s prime. For example, to test 97: √97 ≈ 9.85, so check primes 2, 3, 5, 7. None divide 97 evenly → 97 is prime.
What is prime factorization?
Prime factorization expresses a number as a product of its prime factors. For example: 360 = 2³ × 3² × 5. Every integer greater than 1 has a unique prime factorization (Fundamental Theorem of Arithmetic). It’s used in simplifying fractions, finding GCDs and LCMs, and cryptography.
Are there infinitely many prime numbers?
Yes. Euclid proved this around 300 BC with an elegant argument: assume a finite list of all primes, multiply them together and add 1 — the result is either prime itself or has a prime factor not in your list. Either way, your list was incomplete. Therefore, primes are infinite.