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 is either 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 prime factors:

Where:

• \( n \) — The number to factorize
• \( p_1, p_2, \ldots, p_m \) — 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

Tips: 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 prime factorization of 1?
A: 1 has no prime factors. By definition, prime numbers are greater than 1.

Q2: How long does factorization take for large numbers?
A: This simple calculator uses trial division which becomes slow for numbers >10 digits. More advanced algorithms exist for very large numbers.

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

Q4: What's the largest number this calculator can handle?
A: Practically limited by PHP's integer size (typically 2^63-1 on 64-bit systems). Very large numbers may cause timeouts.

Q5: Can this calculator handle negative numbers?
A: No, it only accepts positive integers ≥ 2. The concept of prime factorization applies to natural numbers.