Prime Number Calculator — Check Primality & Find Factors

How Prime Checker Works

n is prime iff it has exactly two distinct positive divisors: 1 and itself

A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself. To check primality, we use trial division: test whether n is divisible by any integer from 2 up to the square root of n. If none divide evenly, n is prime. This method works because if n has a divisor d greater than sqrt(n), then n/d would be a divisor smaller than sqrt(n), so we only need to check up to sqrt(n).

n

Number to Test — The positive integer being checked for primality. Must be at least 2.

p

Trial Divisor — Each integer we test as a possible divisor of n, starting from 2 and going up to the square root of n.

d(n)

Divisor Count — The total number of positive integers that divide n evenly, including 1 and n. A prime has exactly 2 divisors.

sqrt(n)

Upper Bound for Trial Division — The square root of n. If n has any divisor other than 1 and itself, at least one will be at or below sqrt(n).

How to Use

1. Enter any positive integer between 1 and 10,000,000 in the input field.
2. The calculator instantly tells you whether the number is prime or composite.
3. If the number is composite, view its complete list of prime factors separated by commas.
4. See the divisor count to understand how many numbers divide the input evenly.
5. Read the step-by-step primality check to understand exactly how the determination was made.

Quick Reference

Worked Examples

Frequently Asked Questions

Is 1 a prime number?

No, 1 is not a prime number. By definition, a prime number must have exactly two distinct positive divisors. The number 1 has only one divisor (itself). Excluding 1 from the primes ensures the Fundamental Theorem of Arithmetic holds: every integer has a unique prime factorization. If 1 were considered prime, factorizations would no longer be unique (e.g., 6 = 2 × 3 = 1 × 2 × 3 = 1 × 1 × 2 × 3, etc.).

What is the largest known prime number?

The largest known prime as of early 2025 is 2^136279841 - 1, a Mersenne prime discovered by the Great Internet Mersenne Prime Search (GIMPS) project. It has over 41 million digits. Mersenne primes are of the form 2^p - 1 and are easier to test for primality using the Lucas-Lehmer test. New ones are discovered regularly by distributed computing projects.

What is the difference between a prime and a composite number?

A prime number has exactly two factors: 1 and itself. A composite number has more than two factors — it can be divided evenly by numbers other than 1 and itself. For example, 7 is prime (only 1 and 7 divide it), while 12 is composite (it can be divided by 1, 2, 3, 4, 6, and 12). Every integer greater than 1 is either prime or composite.

How does trial division work for primality testing?

Trial division tests whether n is divisible by any integer d from 2 up to sqrt(n). If n is divisible by any d in this range, then n is composite (we have found a divisor). If no d divides n, then n is prime. We only need to go up to sqrt(n) because if n = a × b and both a and b are greater than sqrt(n), then a × b > n, which is impossible. So at least one factor must be at or below sqrt(n).

How do prime numbers relate to cryptography?

RSA encryption, one of the most widely used encryption systems, relies on the fact that multiplying two large primes together is easy, but factoring the result back into primes is extremely hard. A typical RSA key uses two primes that are each hundreds of digits long. The security of the system depends on this asymmetry — even with the fastest computers, factoring a 2048-bit number into its prime factors would take longer than the age of the universe with current algorithms.

Pro Tips

• →All primes greater than 3 are of the form 6k+1 or 6k-1. Use this fact to quickly spot likely primes: a number not fitting either form cannot be prime (except 2 and 3).
• →Numbers ending in 0, 2, 4, 6, or 8 (even numbers > 2) and numbers ending in 5 (> 5) are always composite — you do not need a calculator for these.
• →If you are testing many numbers, remember that the trial division algorithm checks up to sqrt(n). For n = 10,000,000, this means at most 1,581 odd numbers need to be checked, which is very fast.
• →The divisor count result is especially useful: a prime has exactly 2 divisors, a prime power p^k has k+1 divisors, and composite numbers with many small prime factors can have surprisingly many divisors (e.g., 720 has 30 divisors).
• →For cryptographic or large-number use cases, note that this calculator is limited to 10 million. Prime generation for RSA keys (which uses 1024-bit or 2048-bit numbers) requires specialized libraries and probabilistic primality tests.

Limitations to Know

• •The trial division algorithm becomes computationally expensive for numbers near the 10 million upper bound, though still within practical limits (sqrt(10M) = 3162 divisions).
• •Numbers larger than 10 million cannot be tested with this calculator.
• •The primality test is deterministic for the supported range but does not provide a certificate of primality suitable for cryptographic verification.
• •The prime factorization shown for composite numbers may not be unique in its display format, though the fundamental theorem of arithmetic guarantees uniqueness of the factor set.

Sources

• Wikipedia — Prime Number
• Wolfram MathWorld — Prime Number