Trial Division Algorithm
- Start with the smallest prime (2)
- Divide n by the prime while divisible
- Move to the next prime
- Repeat until n = 1 or √n exceeded
- If n > 1, it's a prime factor
Time Complexity: O(√n) for the optimized method
Prime factorization is the process of breaking down a composite number into a product of its prime factors. According to the Fundamental Theorem of Arithmetic, every integer greater than 1 can be expressed as a unique product of prime numbers (up to the order of factors). This unique representation is extremely useful in mathematics.
For example, 360 = 2³ × 3² × 5, meaning 360 can be written as 2 × 2 × 2 × 3 × 3 × 5. This decomposition is unique - no other combination of primes will multiply to give 360.
Prime factorization has numerous applications in mathematics and computer science:
- Finding GCD and LCM: The greatest common divisor and least common multiple can be easily calculated using prime factorizations.
- Simplifying Fractions: Reduce fractions to lowest terms by canceling common prime factors.
- Cryptography: RSA encryption relies on the difficulty of factoring large numbers.
- Number Theory: Understanding divisibility, perfect numbers, and other properties.
This prime factorization calculator uses deterministic trial division. Very large numbers may require more advanced algorithms (such as Pollard's rho or the quadratic sieve) for faster results. Results are for educational purposes.