Prime Number Checker
Check if any number is prime using an optimized trial division algorithm. Also lists factors.
What Is the Prime Number Checker?
The Prime Number Checker determines whether a given integer is prime — divisible only by 1 and itself. It performs trial division up to the square root of the number, lists all factors if the number is composite, and identifies special prime types (twin prime, Mersenne prime, etc.).
Formula
How to Use
Enter any positive integer greater than 1. The calculator tests primality using trial division and reports: prime or composite. If composite, it shows the complete prime factorization. It also checks if the number is a twin prime, safe prime, or other special category.
Example Calculation
Is 97 prime? Test divisors up to √97 ≈ 9.8: not divisible by 2,3,5,7 → PRIME ✓. Is 91 prime? 91 = 7 × 13 → COMPOSITE. Is 17 prime? Twin prime (17,19), both prime ✓.
Understanding Prime Number Checker
Prime numbers — integers greater than 1 divisible only by 1 and themselves — are the building blocks of all integers. The Fundamental Theorem of Arithmetic guarantees that every integer greater than 1 has a unique prime factorization. This makes primes the atoms of number theory and the foundation of modern cryptography.
Primality testing is a fundamental computational problem. For small numbers, trial division (testing divisors up to √n) is efficient. For large numbers (hundreds of digits), probabilistic tests like Miller-Rabin and deterministic algorithms like AKS are used. RSA encryption relies on the computational difficulty of factoring the product of two large primes.
Despite centuries of study, many prime-related conjectures remain unsolved: the Twin Prime Conjecture (infinitely many primes differ by 2), Goldbach's Conjecture (every even integer > 2 is the sum of two primes), and the Riemann Hypothesis (about the distribution of primes). Primes continue to be at the frontier of both pure mathematics and applied cryptography.
Frequently Asked Questions
Why only test divisors up to √n?
If n has a factor greater than √n, then the corresponding co-factor must be less than √n. So checking all integers up to √n guarantees we find any factor that exists.
What is a twin prime?
Twin primes are pairs of primes that differ by 2: (3,5), (5,7), (11,13), (17,19), (29,31). It is unknown whether infinitely many twin prime pairs exist — the Twin Prime Conjecture remains unproven.
Are there infinitely many prime numbers?
Yes. Euclid proved this around 300 BCE: assume a finite list of primes, multiply them all and add 1. The result is either prime or has a prime factor not on the list — contradiction.
What is prime factorization?
Every integer > 1 can be written as a unique product of prime numbers (Fundamental Theorem of Arithmetic). For example: 360 = 2³ × 3² × 5.
Is this calculator free?
Yes, completely free with no account required.