Prime Number Calculator

By trial division. It divides the number by 2, 3, 5 and then by every candidate on a 2·3·5 wheel up to the square root of the number. Testing beyond the square root is pointless: if n = a × b and both a and b were larger than √n, their product would exceed n. This is an exact test with no probabilistic shortcuts, so numbers that fool quick pseudoprimality checks — Carmichael numbers such as 561 = 3 × 11 × 17 — are correctly reported as composite.

No. A prime has exactly two distinct divisors, and 1 has only one, so it is neither prime nor composite. It also has no prime factorisation at all, which is why this calculator says so in words rather than showing a blank result. The exclusion is not an accident of definition: if 1 were prime, 12 could be written as 2 × 2 × 3, or 1 × 2 × 2 × 3, or 1 × 1 × 2 × 2 × 3, and the fundamental theorem of arithmetic would lose its uniqueness.

2 is prime, and it is the only even prime. Any other even number has 2 as a divisor in addition to 1 and itself, giving it at least three divisors, so it is composite by definition. That makes 2 and 3 the only pair of consecutive primes.

Neither. 0 is divisible by every whole number, so it has no finite divisor list and no prime factorisation — it is neither prime nor composite. Primality is conventionally defined only for integers of 2 and above, so a negative input is rejected here rather than answered, with a message explaining why.

Because the calculator runs in your browser on every keystroke and the answer has to arrive before the next one. Trial division costs roughly √n divisions, and the slowest case is a large prime, where the loop cannot exit early and the nearest-prime search on either side repeats the same full-length work. At 100 billion that whole workload was measured at about 13 ms on average and 19 ms at worst; at a trillion it rose to 40–70 ms, which is visible lag. The cap is where the measurement put it, not a round number.

They say the same thing at different lengths. The exponent form gathers repeated factors together — 360 = 2³ × 3² × 5 — which is the compact notation used in textbooks and the one you want when reducing fractions or finding a lowest common multiple. The expanded form lists every factor separately, 360 = 2 × 2 × 2 × 3 × 3 × 5, which is easier to check by multiplying straight through.