Prime Calculator: Check Primality and Find Prime Factors

Prime numbers are the building blocks of mathematics — integers greater than 1 that are divisible only by 1 and themselves. Our Prime Calculator checks whether any number is prime, finds all prime factors, determines the prime factorization, and generates a list of primes within a given range. Essential for number theory, cryptography, and mathematical exploration.

What Makes a Number Prime?

A prime number has exactly two positive divisors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. The number 2 is the only even prime — all other even numbers are divisible by 2. Numbers with more than two divisors are called composite numbers. The number 1 is neither prime nor composite.

The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be expressed uniquely as a product of primes. This prime factorization is the basis of many cryptographic systems.

Using the Prime Calculator

Enter a number to check its primality. The calculator performs trial division up to the square root of the number, which is the most efficient method for numbers up to about 10 billion. For larger numbers, it uses probabilistic tests based on Fermat's Little Theorem.

You can also use the calculator to generate a list of all primes up to a given limit, find the next prime after a given number, and compute the prime factorization of any composite number showing both the prime factors and their exponents.

Primality Testing Methods

Trial Division

The simplest method: test divisibility by all integers up to the square root of the number. A number is prime if no divisor is found. Efficient for numbers under 10 billion but impractical for very large numbers.

Sieve of Eratosthenes

An ancient algorithm for finding all primes up to a limit. Start with a list of numbers from 2 to the limit. Mark multiples of each prime starting from 2. The unmarked numbers are prime. This is the most efficient method for generating primes up to 10 million.

Probabilistic Tests

For very large numbers (like those used in RSA cryptography), deterministic testing is impractical. Probabilistic tests like Miller-Rabin can determine primality with extremely high confidence by testing random bases.

Applications of Prime Numbers

• RSA encryption: The security of internet encryption relies on the difficulty of factoring the product of two large primes (typically 2048 bits or larger)
• Hash functions: Prime numbers are used in hash table implementations to reduce collisions
• Random number generation: Many PRNGs use prime numbers in their algorithms
• Error-correcting codes: Prime-based Galois fields are used in QR codes, CD/DVD encoding, and satellite communications
• Mathematical research: The Riemann Hypothesis, one of the Millennium Prize Problems, concerns the distribution of prime numbers

Real-World Example

Is 997 a prime number?

• Square root of 997: approximately 31.6
• Test divisibility by primes up to 31: 2 (no), 3 (9+9+7=25 not divisible by 3), 5 (no), 7 (997/7 = 142.4, not integer), 11 (997/11 = 90.6, not integer), 13, 17, 19, 23, 29, 31
• None divide evenly: 997 is prime

What about 999? 999 = 3 × 333 = 3³ × 37, so it is composite. Its prime factorization is 3³ × 37.

The number 1,234,567,890 has prime factorization: 2 × 3 × 3 × 5 × 3,607 × 3,803 = 2 × 3² × 5 × 3,607 × 3,803.

Start Calculating

Use our Prime Calculator below to explore prime numbers and factorization. Also check our GCF and LCM Calculator for greatest common factors and our Combination Calculator for combinatorial number theory.