Prime Number Checker & Factorization

How to Use

Type or paste a positive integer. The tool instantly tells you whether it is prime, shows its prime factorization (for composite numbers), and displays the previous and next prime numbers. The implementation uses 6k ± 1 trial division, which is fast enough to handle numbers well into the trillions without a perceptible delay.

What Is a Prime Number

A prime number is an integer greater than 1 that has no positive divisors other than 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Every integer greater than 1 can be expressed as a unique product of primes — this is the Fundamental Theorem of Arithmetic, and it is the basis for integer factorization algorithms used in cryptography, number theory, and combinatorics.

How the Check Works

The primality test starts by excluding 0, 1, and negative numbers. It handles 2 and 3 directly, then rules out multiples of 2 and 3. Remaining candidates are tested using 6k ± 1 trial division — all primes greater than 3 are of the form 6k−1 or 6k+1, which lets the loop skip two-thirds of potential divisors. Trial division runs up to the square root of n. For factorization, the same pattern extracts prime factors in ascending order.

Common Use Cases

• Students checking homework answers in number theory or discrete math.
• Hobbyists exploring prime sequences, factorial-adjacent patterns, or Euler's totient.
• Teachers preparing worked examples for factorization lessons.
• Quick sanity checks while programming — testing whether a small constant is prime.
• Puzzle solvers and competitive programmers verifying cases.

Tips

• 1 is not prime by convention — primes must have exactly two distinct positive divisors.
• 2 is the only even prime.
• Primes become rarer as numbers grow (Prime Number Theorem: around n/ln(n) primes up to n).
• For cryptography-sized numbers (hundreds of digits), trial division is impractical — probabilistic tests like Miller-Rabin are used instead. This tool is for everyday numbers.
• Input is capped at 14 digits (10^14 − 1) to keep trial division responsive in the browser.

Privacy

All prime-testing and factorization happens in your browser. The numbers you enter are never sent to any server or stored anywhere.