How to Check If a Number Is Prime (2026)

A prime number is a whole number greater than 1 that has exactly two divisors: 1 and itself. Determining whether a number is prime is a foundational problem in mathematics and computer science, with applications in cryptography, hashing, and random number generation.

How primality testing works

The simplest approach is trial division: check whether the number n is divisible by any integer from 2 up to the square root of n. If no divisor is found, the number is prime. You only need to check up to the square root because if n has a factor larger than its square root, the corresponding paired factor must be smaller than it — so you would have already found it. For large numbers, more advanced algorithms like the Miller-Rabin probabilistic test or the AKS deterministic test are used, but trial division is sufficient for numbers up to several million.

Quick reference: prime status of common numbers

Small primes come up frequently in programming, cryptography key sizes, and math puzzles. Here is a reference table for numbers up to 50.

Edge cases to watch for

The number 1 is not prime — a common source of off-by-one errors in implementations. Negative numbers and zero are also not prime by convention. The number 2 is the only even prime; every other even number is divisible by 2 and therefore composite. If you are writing your own check, handling these special cases before entering the trial division loop avoids unnecessary computation and wrong results.

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