1. What is Prime Factorization?

Prime factorization is the process of determining which prime numbers multiply together to create the original number. Every integer greater than 1 either is a prime number or can be represented as a unique product of prime numbers.

2. How Does the Calculator Work?

The calculator uses trial division to find all prime factors:

Where:

• \( n \) — The number to be factorized
• \( p_1, p_2, \ldots, p_m \) — Prime factors
• \( k_1, k_2, \ldots, k_m \) — Their respective exponents

Algorithm: The calculator first divides by 2 until the number is odd, then checks odd divisors up to √n.

3. Importance of Prime Factorization

Applications: Prime factorization is fundamental in number theory, cryptography (RSA algorithm), finding greatest common divisors, and simplifying fractions.

4. Using the Calculator

Instructions: Enter any integer ≥2. The calculator will display its prime factors with exponents for repeated factors.

5. Frequently Asked Questions (FAQ)

Q1: What is the largest number this calculator can handle?
A: It depends on server resources, but numbers up to 14-15 digits should work efficiently.

Q2: Why does 1 not have a prime factorization?
A: By definition, 1 is neither prime nor composite. Prime factorization applies to integers ≥2.

Q3: Are prime factorizations unique?
A: Yes, according to the Fundamental Theorem of Arithmetic, every integer >1 has a unique prime factorization (up to ordering).

Q4: How are repeated factors displayed?
A: Repeated factors are shown with exponents (e.g., 12 = 2² × 3).

Q5: What's the most efficient factorization algorithm?
A: For very large numbers, more sophisticated algorithms like Pollard's Rho are used, but trial division is simplest for moderate numbers.