Prime Factorization Calculator
Find the prime factors of any positive integer. Decompose numbers into their unique prime factorization and discover their divisors.
Understanding Prime Factorization
Definition: Breaking down a number into product of prime numbers
Uniqueness: Every number has a unique prime factorization
Prime Numbers: Numbers with exactly two factors (1 and itself)
Key Properties
Fundamental Theorem: Every integer > 1 has a unique prime factorization
Divisors: Can be found using prime factors
Perfect Numbers: Equal to sum of their proper divisors
Fundamental Theorem of Arithmetic
The prime factorization of integers represents one of the most profound concepts in number theory, encapsulated in the Fundamental Theorem of Arithmetic. This theorem establishes that every positive integer greater than 1 can be represented uniquely as a product of prime powers. The mathematical significance of this theorem extends beyond simple factorization, forming the foundation for numerous results in advanced number theory and abstract algebra.
The uniqueness property of prime factorization provides a canonical form for representing integers, enabling systematic analysis of their multiplicative structure. This representation reveals deep connections between different numbers through their prime factor patterns and exponent relationships.
Mathematical Framework
The formal expression of prime factorization follows specific mathematical notation:
n = p1^a1 × p2^a2 × ... × pk^ak
Where:
- p1 < p2 < ... < pk are prime numbers
- a1, a2, ..., ak are positive integers
- k is the number of distinct prime factors
Key Properties:
- ω(n) = number of distinct prime factors
- Ω(n) = total number of prime factors
- rad(n) = p1 × p2 × ... × pk
Divisor Theory
The prime factorization enables systematic analysis of divisors through combinatorial methods. The number and sum of divisors can be expressed through prime factor exponents:
Number of Divisors: d(n) = Π(ai + 1)
Sum of Divisors: σ(n) = Π((pi^(ai+1) - 1)/(pi - 1))
Perfect Number Condition:
σ(n) - n = n (Perfect)
σ(n) - n > n (Abundant)
σ(n) - n < n (Deficient)
Algorithmic Complexity
The computational aspects of prime factorization involve sophisticated algorithms with varying complexity characteristics. The fundamental algorithms include:
Trial Division:
- • Time Complexity: O(√n)
- • Space Complexity: O(log n)
- • Optimal for small numbers
Advanced Methods:
- • Pollard's Rho: O(n^(1/4))
- • Quadratic Sieve: exp(√(log n × log log n))
- • Number Field Sieve: exp((log n)^(1/3) × (log log n)^(2/3))
Number Theoretic Functions
Prime factorization enables the computation of important number theoretic functions that reveal deeper properties of integers:
Euler Totient: φ(n) = n × Π(1 - 1/p)
Möbius Function: μ(n) = (-1)^ω(n) if square-free
Liouville Function: λ(n) = (-1)^Ω(n)
Properties:
- Multiplicative: f(mn) = f(m)f(n) if gcd(m,n) = 1
- Completely multiplicative: f(mn) = f(m)f(n) for all m,n