Prime Number Checker

Unsupported input

The tool may reject input that does not match the expected content, structure, or file type.

Fix: Confirm the tool input requirements and paste the correct type of data.

Incomplete values

Missing fields or partial content can block processing or produce weak results.

Fix: Provide the full required input before running the tool.

Copying placeholder content

Sample or placeholder values can lead to output that looks valid but is not ready for real use.

Fix: Replace placeholders with your actual values before relying on the result.

Is the Miller–Rabin test good enough for cryptography?

For key-sized primes (1024+ bits), production cryptography libraries use Miller–Rabin with 40+ rounds, giving a false-positive rate below 2^-80. This tool uses 12 witness bases, which is deterministic up to 3.3 × 10^14 and delivers ~2^-36 false-positive above that — enough for study and verification, but you should use a proper library (OpenSSL, BoringSSL) for actual key generation.

How big a number can the tool handle?

Up to 2^64 − 1 (18,446,744,073,709,551,615). Above that, primality testing needs arbitrary-precision arithmetic (BigInt), which would slow the browser down for the test frequencies we expect. For 1024-bit or larger primes, use a CAS like WolframAlpha or a scripting environment with a GMP binding.

What's the difference between trial division and Miller–Rabin?

Trial division tries every prime up to √n as a divisor — exact but slow beyond 10^9. Miller–Rabin tests a few randomized witnesses; for n ≥ 2, if any witness 'fails', n is definitely composite. With enough witnesses, the probability of missing a composite is negligible. Miller–Rabin is how real-world primality testing is done.

Does 'Composite' output show all prime factors?

Yes. The factorization output lists every prime factor and its multiplicity. For example, 360 returns 'Composite — 2³ × 3² × 5'. For very large composites with no small factors, the tool uses Pollard's rho to extract factors before handing off to Miller–Rabin on each factor.

How is the Sieve of Eratosthenes implemented?

Standard sieve with odd-only optimization: a boolean array of length n/2 is initialized, then for each prime p up to √n, all odd multiples of p are marked composite. The sieve handles an upper bound of 10,000,000 in roughly 200 ms on a modern browser — memory cost scales linearly with the bound.

Why is 1 not considered prime?

By definition, a prime has exactly two distinct divisors: 1 and itself. 1 has only one divisor (itself), so it doesn't meet the definition. Excluding 1 is also the only convention that makes the Fundamental Theorem of Arithmetic — unique prime factorization — work without special cases. The tool returns 'Not prime' for 1 and 0, and 'Invalid' for negatives.