Prime Number Checker
Check if a number is prime and find its factors.
What is a Prime Number?
A prime number is a natural number greater than 1 that has exactly two factors: 1 and itself. The first primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47...
Key facts: 2 is the only even prime. 1 is not prime (by modern convention). There are infinitely many primes (proven by Euclid around 300 BC). As numbers get larger, primes become less frequent but never stop appearing.
How to Check if a Number is Prime
Trial division: test whether any integer from 2 to sqrt(n) divides n evenly. If none do, n is prime. You only need to check up to the square root because if n = a × b, at least one of a or b must be less than or equal to sqrt(n).
Optimization: after checking 2, you only need to check odd numbers (3, 5, 7...). Further: check 2, 3, then only numbers of the form 6k±1 (since all primes greater than 3 are of this form). This reduces checks by 2/3.
Why Primes Matter
Primes are the building blocks of all integers — every number can be uniquely expressed as a product of primes (Fundamental Theorem of Arithmetic). 60 = 2² × 3 × 5.
Modern applications: Cryptography (RSA encryption relies on the difficulty of factoring products of large primes), hash functions (prime-sized tables reduce collisions), pseudorandom number generators, and error-correcting codes. Internet security fundamentally depends on prime numbers.
Frequently Asked Questions
Is 1 a prime number?
No. By modern mathematical convention, 1 is neither prime nor composite. Excluding 1 from primes preserves the uniqueness of prime factorization (the Fundamental Theorem of Arithmetic). Historically, some mathematicians did consider 1 prime, but the modern definition excludes it.
What is the largest known prime?
As of 2024, the largest known prime is 2^136,279,841 − 1 (a Mersenne prime), discovered in October 2024. It has over 41 million digits. The Great Internet Mersenne Prime Search (GIMPS) project finds most record primes using distributed computing.
Are there patterns in prime numbers?
Primes appear somewhat random, but patterns exist. Twin primes (differing by 2, like 11 and 13) appear to continue forever (unproven). The Prime Number Theorem states that primes near N occur with frequency approximately 1/ln(N). The Riemann Hypothesis — one of the greatest unsolved problems in mathematics — concerns the precise distribution of primes.