Prime Number Checker & Factorizer
Check if any positive integer is prime. If it is not prime, see its complete prime factorization tree. Also find the next 10 primes after your number. 100% private — runs in your browser.
What Are Prime Numbers?
A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. In other words, a prime has exactly two factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. The number 2 is special because it is the only even prime number — every other even number is divisible by 2 and therefore not prime.
Numbers greater than 1 that are not prime are called composite numbers. Every composite number can be expressed as a unique product of prime factors — this is known as the Fundamental Theorem of Arithmetic. For example, 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7.
Prime Checking Algorithm
To check if n is prime, test divisibility by all integers from 2 to √n
Why only up to √n? Because if n = a × b and both a and b are greater than √n, then a × b > n, which is a contradiction. So at least one factor must be ≤ √n.
The Fundamental Theorem of Arithmetic
Every integer greater than 1 is either a prime itself or can be represented as a product of prime numbers in exactly one way (ignoring the order of factors). This uniqueness of prime factorization is one of the most fundamental results in number theory and underpins much of modern mathematics.
Primes in Cryptography
Prime numbers are the backbone of modern cryptography. RSA encryption, which secures most internet communication, relies on the difficulty of factoring the product of two large primes. While it is easy to multiply two 300-digit primes together, no known algorithm can efficiently factor the resulting 600-digit number. This asymmetry between multiplication and factorization is what makes your online banking, emails, and messages secure.
Prime Number Patterns
Despite centuries of study, primes remain mysterious. The Prime Number Theorem tells us that primes become less frequent as numbers get larger, roughly following the pattern n/ln(n). Twin primes (pairs like 11,13 or 29,31 that differ by 2) are conjectured to be infinite, but this has never been proven. The Riemann Hypothesis, one of the greatest unsolved problems in mathematics, is intimately connected to the distribution of prime numbers.
Common Uses for Prime Factorization
Beyond cryptography, prime factorization is used to simplify fractions (find GCD), calculate LCM for scheduling problems, solve Diophantine equations, and analyze number patterns. In computer science, hash functions often use prime numbers to reduce collisions, and prime-sized hash tables distribute keys more uniformly.