Prime Factorization
Decompose a positive integer into its unique prime factor representation. Every integer greater than 1 can be expressed as a product of prime numbers in exactly one way (Fundamental Theorem of Arithmetic). This algorithm uses trial division optimized to check only up to √n, as any composite number must have a prime factor ≤ √n. Returns a map of prime factors to their powers, e.g., 360 = 2³ × 3² × 5¹.
Visualization
Interactive visualization for Prime Factorization
Interactive visualization with step-by-step execution
Implementation
1function primeFactorization(n: number): Map<number, number> {
2 const factors = new Map<number, number>();
3
4 // Check for 2
5 while (n % 2 === 0) {
6 factors.set(2, (factors.get(2) || 0) + 1);
7 n /= 2;
8 }
9
10 // Check odd numbers from 3 onwards
11 for (let i = 3; i * i <= n; i += 2) {
12 while (n % i === 0) {
13 factors.set(i, (factors.get(i) || 0) + 1);
14 n /= i;
15 }
16 }
17
18 // If n is still > 1, it's a prime factor
19 if (n > 1) {
20 factors.set(n, 1);
21 }
22
23 return factors;
24}Deep Dive
Theoretical Foundation
The Fundamental Theorem of Arithmetic guarantees that every integer n > 1 has a unique prime factorization. The trial division method checks divisibility starting from 2, then odd numbers from 3 onwards. The key optimization is stopping at √n because if n has a factor greater than √n, it must have a corresponding factor less than √n. We repeatedly divide by each factor until it no longer divides n, counting the multiplicity. Time complexity is O(√n) worst case when n is prime.
Complexity
Time
O(log n)
O(√n)
O(√n)
Space
O(log n)
Applications
Industry Use
RSA cryptography (factoring semiprime for breaking)
Computing totient function φ(n)
Finding all divisors of a number
Simplifying radicals in mathematics
GCD and LCM computation via prime factors
Perfect number identification
Cryptanalysis and security testing
Number theory proofs and problems
Pollard's p-1 algorithm preprocessing
Use Cases
Related Algorithms
GCD (Euclidean Algorithm)
Compute the Greatest Common Divisor (GCD) of two integers using the Euclidean algorithm. Dating back to around 300 BC and appearing in Euclid's Elements, it's one of the oldest algorithms still in common use. The algorithm is based on the principle that GCD(a,b) = GCD(b, a mod b) and is remarkably efficient with O(log min(a,b)) time complexity. The GCD is the largest positive integer that divides both numbers without remainder.
LCM (Least Common Multiple)
Calculate the Least Common Multiple (LCM) of two integers - the smallest positive integer that is divisible by both numbers. The LCM is intimately related to the GCD through the formula: LCM(a,b) = |a×b| / GCD(a,b). This relationship allows us to compute LCM efficiently using the Euclidean algorithm for GCD, achieving O(log min(a,b)) time complexity instead of naive factorization methods.
Sieve of Eratosthenes
Ancient and highly efficient algorithm to find all prime numbers up to a given limit n. Invented by Greek mathematician Eratosthenes of Cyrene (276-194 BC), this sieve method systematically eliminates multiples of primes, leaving only primes in the array. With O(n log log n) time complexity, it remains one of the most practical algorithms for generating large lists of primes, vastly superior to trial division which runs in O(n² / log n) time.
Modular Exponentiation (Fast Power with Modulo)
Efficiently compute (base^exponent) mod m without overflow, crucial for cryptographic operations. Computing large powers like 2^100 mod 1000000007 is impossible with naive exponentiation due to integer overflow, but modular exponentiation achieves this in O(log exponent) time using the binary representation of the exponent. This algorithm is the foundation of RSA encryption, Diffie-Hellman key exchange, and many cryptographic protocols.