First 20 Prime Numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
31, 37, 41, 43, 47, 53, 59, 61, 67, 71
A number n is prime if no integer from 2 to √n divides it evenly.
Prime Number Checker
Check if any number is prime, find its prime factors, or generate all prime numbers in a given range. A prime number is a natural number greater than 1 that has no divisors other than 1 and itself.
Prime Number Checker
Frequently Asked Questions
Is 1 a prime number?
No. By definition, prime numbers must have exactly two distinct factors: 1 and themselves. The number 1 has only one factor (itself), so it's neither prime nor composite. This exclusion is important for the fundamental theorem of arithmetic (unique prime factorization).
What is prime factorization used for?
Prime factorization breaks a number into its prime building blocks. It's used to find greatest common divisors (GCD), least common multiples (LCM), simplify fractions, and is the foundation of RSA encryption — the security of which relies on the difficulty of factoring large numbers.
What makes a number prime
A prime has exactly two distinct divisors: 1 and itself. 7 qualifies; 9 does not, because 3 divides it.
1 is not prime — it has only one divisor. This is not an arbitrary exclusion: treating 1 as prime would break the fundamental theorem of arithmetic, which states every integer above 1 factors into primes in exactly one way. If 1 were prime, 6 could be 2×3 or 1×2×3 or 1×1×2×3, and uniqueness would collapse.
Testing by hand
Trial division only needs to run up to the square root of the number. If no prime at or below √n divides n, then n is prime.
The reason is that factors come in pairs: if n = a × b, one of them must be at or below √n. Checking beyond that point only re-finds partners you have already tested.
Example: for 97, √97 ≈ 9.8, so test 2, 3, 5 and 7. None divide it, so 97 is prime.
Quick divisibility checks
- 2 — last digit is even
- 3 — digit sum divisible by 3
- 5 — ends in 0 or 5
- 11 — alternating digit sum divisible by 11
Why primes matter
Modern public-key cryptography rests on an asymmetry: multiplying two large primes is fast, while factoring the product back into those primes is computationally infeasible at sufficient size. The security of a great deal of internet traffic depends on that gap.
Primes thin out as numbers grow but never run out — Euclid proved there are infinitely many, by a short argument still taught today.