Prime Number Checker

A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. Our prime number checker uses efficient algorithms to determine whether a given number meets this criterion by testing its divisibility against potential factors.



For smaller numbers (typically up to several million), the checker employs trial division, which tests divisibility by all numbers from 2 up to the square root of the input number. This optimization dramatically reduces computation time—for example, to check if 10,007 is prime, we only need to test divisors up to 100 rather than all 10,006 potential divisors. The square root boundary works because if a number n has a divisor larger than √n, it must also have a corresponding divisor smaller than √n.



For very large numbers, the tool can implement probabilistic primality tests like the Miller-Rabin algorithm, which provides extremely high confidence (though not absolute certainty) that a number is prime. These tests are used in cryptography for generating large prime numbers for RSA encryption. When a number is determined to be composite (not prime), the tool provides its prime factorization, breaking it down into the prime numbers that multiply together to create it. This factorization is unique for every number according to the Fundamental Theorem of Arithmetic.



All calculations happen entirely in your browser using JavaScript's number handling and big integer libraries for larger values. This client-side approach ensures privacy—your numbers never leave your device—and provides instant results without server communication delays.

1. Mathematics Education & Learning
Students learning number theory use prime checkers to verify homework answers, explore prime patterns, and understand the distinction between prime and composite numbers. Teachers create exercises around prime identification, and students can immediately verify their manual calculations. The tool helps illustrate concepts like the Sieve of Eratosthenes, Goldbach's Conjecture (every even number > 2 is the sum of two primes), and twin primes (primes differing by 2, like 11 and 13).



2. Cryptography & Computer Security
Cryptographic systems, particularly RSA encryption, rely on the difficulty of factoring large composite numbers created from two huge primes. Security professionals and computer science students use prime checkers to understand key generation, verify that randomly generated large numbers are indeed prime, and explore the computational complexity that makes modern encryption secure. Testing primality of 100+ digit numbers demonstrates why factoring is computationally infeasible.



3. Programming & Algorithm Development
Software developers implementing prime-related algorithms (Sieve of Eratosthenes, prime factorization engines, cryptographic libraries) use prime checkers to verify their code's correctness. They test edge cases (2 is the only even prime, 1 is not prime), performance benchmarks, and correctness across ranges. The tool serves as a reference implementation for validating custom primality testing functions.



4. Mathematical Research & Pattern Exploration
Mathematicians and enthusiasts explore unsolved problems in number theory: Are there infinitely many twin primes? What's the distribution of primes (addressed by the Prime Number Theorem)? Mersenne primes (2ⁿ - 1) and their connection to perfect numbers? Testing sequences of numbers for primality reveals patterns and gaps, supporting conjectures or discovering counterexamples.



5. Competitive Mathematics & Olympiads
Students preparing for math competitions (AMC, AIME, IMO) frequently encounter prime number problems. Quick primality testing helps verify answers during practice, explore properties of specific numbers, and understand relationships between primes and other number-theoretic concepts like GCD, LCM, modular arithmetic, and Fermat's Little Theorem (if p is prime and a not divisible by p, then a^(p-1) ≡ 1 mod p).

• Remember 2 is the only even prime: Every even number greater than 2 is divisible by 2, making it composite. This makes 2 unique—the smallest and only even prime number.



• Quick divisibility rules: Before using the tool, apply basic divisibility tests: if last digit is 0, 2, 4, 6, 8, divisible by 2; if digits sum to multiple of 3, divisible by 3; if ends in 0 or 5, divisible by 5.



• Test up to the square root: When manually checking primality, only test divisors up to √n. For n=121, only check up to 11 (11×11=121), discovering 121 is composite.



• Understand factorization uniqueness: Every composite number has exactly one prime factorization (order doesn't matter). 60 = 2²×3×5, never any other combination of primes.



• Explore prime patterns: Notice prime gaps increase as numbers grow (primes become less frequent). Between 1-100: 25 primes. Between 1000-1100: only 16 primes.



• Use for RSA key generation understanding: Generate two large primes (e.g., 100+ digits), multiply them, and understand why factoring the product back to the original primes is computationally infeasible.



• Check Mersenne primes: Test numbers of form 2ⁿ-1. When n is prime, 2ⁿ-1 might be prime (Mersenne prime). Examples: 2³-1=7, 2⁵-1=31, 2⁷-1=127 are prime, but 2¹¹-1=2047=23×89 is not.



• Verify twin primes: Find pairs of primes differing by 2: (3,5), (5,7), (11,13), (17,19), (29,31), (41,43). Twin Prime Conjecture (unproven) states infinitely many exist.



• Apply to GCD/LCM problems: When two numbers are coprime (GCD=1), they share no prime factors. Use prime factorization to find GCD (product of common prime factors) and LCM (product of all prime factors at highest powers).



• Understand probabilistic vs deterministic tests: For very large numbers, deterministic tests may be slow. Probabilistic tests (Miller-Rabin) provide 99.99%+ confidence much faster, acceptable for most applications.