Prime Number Checker

Prime Number Checker

Check a Single Number

List All Primes Up to N (max 10,000)

Prime Factorization

Prime Number Checker – Complete Guide

A prime number is a natural number greater than 1 that has exactly two positive divisors: 1 and itself. Examples include 2, 3, 5, 7, 11, 13, 17, 19, and 23. The number 1 is not considered prime by modern definition. Our free online prime number checker tells you instantly whether any number is prime, lists all primes up to a given limit, and shows the full prime factorization of any number.

How to Check if a Number is Prime

To check primality, you test whether the number is divisible by any integer from 2 up to its square root. If no divisor is found, the number is prime. This calculator uses an efficient trial division algorithm for single numbers and the Sieve of Eratosthenes for generating lists of primes.

The Sieve of Eratosthenes

The Sieve of Eratosthenes, developed by the ancient Greek mathematician Eratosthenes around 240 BC, is one of the oldest and most efficient methods for finding all primes up to a given limit. The algorithm works by starting with 2, marking all of its multiples as composite, then moving to the next unmarked number and repeating the process. What remains unmarked are the prime numbers.

This calculator uses a modern implementation that can find all primes up to 10,000 in milliseconds.

Prime Factorization

Every integer greater than 1 is either prime itself or can be expressed as a unique product of prime numbers — a fact known as the Fundamental Theorem of Arithmetic. For example, 360 = 2³ × 3² × 5. This factorization is unique and is the foundation of much of number theory, including cryptographic algorithms like RSA encryption.

Interesting Prime Facts

  • 2 is the only even prime number.
  • There are infinitely many prime numbers (proved by Euclid around 300 BC).
  • The largest known prime (as of 2024) has over 41 million digits.
  • Twin primes are pairs differing by 2, like (11, 13) and (17, 19).
  • Primes are fundamental to modern encryption (RSA, elliptic curve cryptography).

Frequently Asked Questions

Is 1 a prime number?
No. By the modern definition, prime numbers must have exactly two distinct positive divisors. The number 1 has only one divisor (itself), so it is classified as neither prime nor composite.
Is 2 a prime number?
Yes, 2 is prime and it is the only even prime number. All other even numbers are divisible by 2, so they have at least three divisors and cannot be prime.
How large a number can this checker handle?
The single-number primality check works for numbers up to about 10¹² (one trillion) within browser limits. The list feature is capped at 10,000 for performance. The factorization tool also works up to 10¹².
Why are primes important in cryptography?
Multiplying two large primes together is fast, but factoring the result back into its prime components is computationally infeasible for very large numbers. This asymmetry is the security basis of RSA encryption, which protects online banking, HTTPS, and email.
What is prime factorization used for?
Prime factorization is used to find GCD and LCM, simplify fractions, solve modular arithmetic problems, and in cryptography. It is also a standard topic in school number theory curricula.