Free Prime Number Calculator - Check Primes Instantly

Discover the properties of any number with our comprehensive prime number calculator. Whether you need to check if a specific number is prime, find its prime factorization, or generate lists of prime numbers, our tool provides instant, accurate results for educational and research purposes.

Perfect for students learning number theory, mathematicians researching prime patterns, and anyone curious about the fundamental building blocks of numbers. Our calculator handles numbers up to 1 million with lightning-fast algorithms.

Check if a Number is Prime

Enter any positive integer up to 1,000,000

Understanding Prime Numbers: A Complete Guide

Prime vs Composite Numbers

Prime Numbers

  • • Have exactly 2 factors: 1 and itself
  • • Cannot be divided evenly by any other number
  • • Examples: 2, 3, 5, 7, 11, 13, 17, 19
  • • Building blocks of all integers

Composite Numbers

  • • Have more than 2 factors
  • • Can be divided evenly by numbers other than 1 and itself
  • • Examples: 4, 6, 8, 9, 10, 12, 14, 15
  • • Can be factored into prime numbers

Special Cases and Rules

Number 1

The number 1 is neither prime nor composite. It has only one factor (itself), so it doesn't meet the definition of a prime number (which must have exactly 2 factors).

Number 2

The number 2 is the only even prime number. All other even numbers are divisible by 2, making them composite.

Infinity of Primes

There are infinitely many prime numbers. This was proven by Euclid around 300 BCE using an elegant mathematical proof.

Methods for Finding Prime Numbers

Trial Division

Test divisibility by all numbers from 2 up to the square root of the number. If no divisors are found, the number is prime.

Most straightforward method, used by our calculator for accuracy.

Sieve of Eratosthenes

Ancient algorithm that finds all primes up to a given limit by iteratively marking multiples of each prime as composite.

Efficient for finding all primes in a range.

Probabilistic Tests

Advanced methods like Miller-Rabin test for very large numbers. Provide probabilistic results with high confidence.

Used in cryptography for large prime generation.

Famous Prime Numbers and Patterns

Special Prime Types

Mersenne Primes

Primes of the form 2ⁿ - 1. Examples: 3, 7, 31, 127

Used in the search for the largest known primes.

Twin Primes

Prime pairs that differ by 2. Examples: (3,5), (5,7), (11,13)

Subject of the famous Twin Prime Conjecture.

Palindromic Primes

Primes that read the same forwards and backwards: 11, 101, 131

Intersection of number patterns and prime properties.

Prime Number Theorems

Prime Number Theorem

Describes the asymptotic distribution of primes. The probability that a random number n is prime is approximately 1/ln(n).

Goldbach's Conjecture

Every even integer greater than 2 can be expressed as the sum of two primes. Still unproven but verified for large numbers.

Riemann Hypothesis

One of the most famous unsolved problems in mathematics, relating to the distribution of prime numbers.