Prime Number Checker
Enter any positive integer and instantly find out if it's prime. Get prime factorization, all divisors, nearest prime numbers, and a helpful breakdown — all in your browser, no signup required.
Enter a Number
Enter a number and click Check Number to see if it's prime, along with its factorization and nearest primes.
What does this prime number checker do?
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. In other words, a prime number can't be divided evenly by any other number except 1 and itself. The number 7 is prime because the only way to get 7 by multiplication is 1 × 7. But 8 is not prime because it can be written as 2 × 4.
This free online prime number checker instantly tells you whether any positive integer is prime or composite. But it does much more than a simple yes/no — it also shows you the prime factorization, lists all divisors, finds the nearest prime numbers (both above and below), and tells you useful properties like whether the number is a twin prime or a Mersenne prime.
Everything runs locally in your browser. The number you enter never leaves your device — no server, no logs, no tracking. You can check as many numbers as you want, with no limits and no signup.
How to check if a number is prime
- Enter a number: Type any positive integer into the input field. You can use up to 15 digits.
- Click Check Number: The tool instantly runs a primality test using trial division optimized with the 6k±1 rule.
- Read the verdict: A large, colour-coded result tells you whether the number is prime or composite.
- Explore the details: Below the verdict, you'll find the prime factorization (if composite), all divisors, and the nearest prime numbers.
- Copy if needed: Use the "Copy Results" button to grab a formatted summary for homework, reports, or notes.
That's it. No signup, no email, no waiting. The entire process takes seconds.
How the primality test works
This calculator uses trial division with the 6k±1 optimization — a method that's both fast and simple enough to run instantly in your browser.
Step 1: Handle small cases
Numbers less than 2 are neither prime nor composite. 2 and 3 are prime by definition. Any even number greater than 2 is automatically composite (divisible by 2). Any number divisible by 3 is also composite.
Step 2: Test divisibility up to √n
To check if a number n is prime, we only need to test divisors up to the square root of n. This is because if n has a factor larger than √n, it must also have a corresponding factor smaller than √n. For example, to check if 97 is prime, we only test divisors up to √97 ≈ 9.8 — that's just 2, 3, 5, and 7.
Step 3: Use the 6k±1 rule
Every prime number greater than 3 can be written in the form 6k ± 1, where k is some integer. This means we only need to test numbers of the form 6k−1 and 6k+1 (like 5, 7, 11, 13, 17, 19, 23, 25, ...). This skips about two-thirds of all possible divisors, making the test much faster for large numbers.
i = 5 while i * i <= n: if n % i == 0 or n % (i + 2) == 0: return "composite" i = i + 6 return "prime"
This approach is fast enough to check numbers with up to 15 digits in milliseconds, which covers the vast majority of everyday use cases — homework, coding practice, cryptography basics, and number theory exploration.
Key properties of prime numbers
- 2 is the only even prime. Every other even number is divisible by 2, so it can't be prime.
- 1 is not prime. By definition, a prime number must have exactly two distinct positive divisors. The number 1 has only one divisor (itself).
- Every integer greater than 1 is either prime or composite. There's no third category.
- Every composite number has a unique prime factorization. This is the Fundamental Theorem of Arithmetic — you can write any composite number as a product of primes in exactly one way (ignoring the order).
- There are infinitely many primes. Euclid proved this over 2,000 years ago — there's no largest prime number.
- Primes get rarer as numbers get larger. About 1 in ln(n) numbers near n are prime. For example, around 1,000, about 1 in 7 numbers is prime; around 1,000,000, it's about 1 in 14.
- Twin primes are pairs of primes that differ by 2, like (3, 5), (11, 13), and (41, 43). The twin prime conjecture says there are infinitely many such pairs, but nobody has proven it yet.
- Mersenne primes are primes of the form 2p − 1, where p is also prime. The largest known prime is a Mersenne prime with over 41 million digits.
The first 100 prime numbers
Here's a quick reference list of the first 100 primes. Prime numbers are highlighted in green; numbers that aren't prime are shown in grey for context.
Why prime numbers matter
Cryptography
RSA encryption — which secures most internet traffic, banking, and messaging — relies on the fact that multiplying two large primes is easy, but factoring the product back into primes is computationally very hard.
Hash functions
Prime numbers are used in hash tables and hash functions to reduce collisions and distribute data more evenly across buckets.
Random number generation
Prime numbers appear in pseudorandom number generators, which are used in simulations, games, and statistical sampling.
Music theory
Prime numbers show up in rhythmic patterns and polyrhythms — composers like Xenakis used them to create complex, non-repeating musical structures.
Cicadas
Some cicada species have prime-numbered life cycles (13 or 17 years) to avoid syncing with predators that have shorter, non-prime cycles.
Coding interviews
Prime checking is one of the most common coding interview questions. This tool helps you verify your algorithm's output quickly.