How to Check a Prime Number

Our prime number checker tests one whole number up to 1e12 by trial division. If the number is composite it shows the smallest factor greater than 1. It does not list primes in a range or generate keys. For every divisor, use the factor calculator.

Enter an integer. 97 should read Prime. 91 should read Not prime with factor 7. 0 and 1 read not prime.

You never need a factor larger than √n. If n is even and greater than 2, stop. Then try 3, 5, 7… until the square root. That is the method the tool uses.

√97 ≈ 9.8, so testing 2, 3, 5, 7 is enough. √91 ≈ 9.5, and 7 divides 91.

The checker stops at the first factor. The factor calculator lists every divisor and writes n as p^a × q^b. Factorial n! is unrelated — that lives on the scientific calculator.

1 fails the two-divisor rule, so the prime number checker labels it not prime before any trial division. 91 looks prime (odd, not a multiple of 5) until 7×13. 121 is 11² — you only need to test through √121 = 11. Even numbers greater than 2 are finished on the first step: the smallest factor is 2. That is the entire product people want from “prime number checker”: yes or no, and a factor when the answer is no. Generating a list from 1 to 100, proving a 300-digit probable prime, or writing n! are different jobs. The factor calculator then writes the unique prime factorization the fundamental theorem promises for every n > 1.