Prime Numbers

Features

• Five modes: test, factorise, Nth prime, range and random generation
• Miller-Rabin testing: milliseconds even at eighteen digits, with no trial division to the root
• Verdicts below 3.3×10²⁴ are deterministic; above that the tool marks them probabilistic
• Prime factorisation via Pollard rho, which splits the product of two large primes
• A failing test reports the smallest prime factor so you can check it
• Factorisation lists each prime with its exponent (`2^3 × 3^2 × 5`), each copyable
• Carmichael numbers — the composites that fool Fermat’s little theorem — are caught
• Ranges list up to 500 primes, and say how many were left out
• Everything runs on your machine with no external service

How to use

1. Choose a mode: test, factorise, Nth prime, range or random
2. Fill in the corresponding value (range and random modes need a range)
3. The result appears as you type; copy a single row with its button

FAQ

What does “probabilistic” mean here? Can it be wrong?

Miller-Rabin is probabilistic, but with a fixed set of bases it has no counterexamples within a bounded range — it is proven that with the first twelve primes (2 to 37) as bases, no composite below 3.3×10²⁴ can pass as prime. So within that range the verdict is deterministic. Above it the test is still extremely reliable (error probability below 4⁻¹²), it is just no longer a proof, and the tool says so.

How large a number can be factorised?

The limit is eighteen digits. Factorisation combines trial division by small primes, Pollard rho and Miller-Rabin, and the product of two large eighteen-digit primes (about 60 bits) usually resolves within a second. Past that the tool refuses and explains, rather than spinning for minutes — a page that stops responding is far worse than a sentence saying the number is too big.

Why is the first prime 2 rather than 1?

Because 1 is not prime by definition (a prime has exactly two positive divisors and 1 has only one). So the first prime is 2 and the sequence runs 2, 3, 5, 7, 11… This tool also says plainly that 0 and 1 are not prime, and does not add a note about their “smallest prime factor” — there is none, and such a note would be wrong.