Test primality using Miller-Rabin algorithm. Generate large primes for cryptographic key generation.
Paste any integer and the tool answers yes or no: is it prime? It runs the Miller-Rabin test, the algorithm behind RSA and Diffie-Hellman key generation. Give it a bit length and it also produces a random prime of that size. Two things surprise users. For large numbers the verdict is probabilistic: each extra round shrinks the worst-case error by a factor of four, and only small numbers get a definitive answer. And testing primality is not factoring: a composite result comes back without its factors.
A concrete input and expected output from the current implementation.
Input
1000000007
Expected output
Prime
1,000,000,007 is the well-known 10^9 + 7 prime used as a modulus in competitive programming. Because the number sits far below 3.47 trillion, the fixed-base version of Miller-Rabin is exact here: the answer is a proof, not a probability.
Miller-Rabin writes n - 1 as 2^s times an odd number d, then computes a^d mod n and its successive squares. The number passes the round for base a if a^d is 1 mod n or one of those squares is n - 1; otherwise n is composite, with certainty. Each additional random base multiplies the chance of wrongly passing by at most 1/4.
Below 3,474,749,660,383 the bases 2, 3, 5, 7, 11, 13, 17 are proven sufficient, so small inputs get an exact verdict, and the seven-base set 2, 325, 9375, 28178, 450775, 9780504, 1795265022 is verified exact for every number under 2^64. For generation, the tool draws random odd candidates of the requested bit length and retests until one passes; only about one in every 710 numbers near 2^1024 is prime.
For small numbers the test is exact and returns a plain verdict. Above a few trillion, no small fixed base set is proven, so the honest output is 'probably prime' with an error bound: k rounds cap the false-positive rate at 4^-k, and 20 rounds is about one in a trillion. Standards such as FIPS 186-5 accept this and specify how many rounds each bit length requires.
No. Primality testing is a yes-or-no question and stays fast even for thousands of digits, whereas factoring is deliberately hard, which is exactly what RSA security rests on. The largest RSA modulus factored to date is RSA-250, a 250-digit number, and a 2048-bit modulus of 617 digits is far beyond known methods. For factors, use a dedicated factoring tool.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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