Introduction

Project Euler's Problem 3, often referred to as the "Largest Prime Factor" problem, is a classic problem in algorithmic mathematics. It tasks you with finding the largest prime factor of a given composite number. While seemingly straightforward, this problem introduces several programming concepts, including prime factorization and efficient algorithm design.

Problem Statement

The problem is defined as follows:

› What is the largest prime factor of the number 600851475143?

Approach

Solving this problem effectively requires an understanding of both number theory and algorithmic efficiency. Here, we'll explore an optimal solution using mathematical concepts and programming techniques.

Basic Definitions

1. Prime Number: A natural number greater than 1 that has no positive divisors other than 1 and itself.
2. Prime Factor: A factor that is a prime number.
3. Composite Number: A natural number greater than 1 that is not prime, i.e., it has factors other than 1 and itself.

Prime Factorization

The key to solving this problem efficiently is through prime factorization. Prime factorization involves decomposing a number into a product of prime numbers. For instance, the number 28 can be factorized into 2, 2, and 7.

Example

To prime factorize a number:

1. Start with the smallest prime number, 2.
2. Divide the number by this prime if it is divisible.
3. Repeat the process with the quotient until it equals 1.
4. Move to the next prime number if the quotient is not divisible by the current prime.

Optimal Solution

The solution involves iterating through potential factors starting from the smallest prime, 2, and progressively eliminating them.

Implementation Details

Here's a step-by-step breakdown of the algorithm:

1. Initialize the dividend as the target number, 600851475143.
2. Start with the smallest prime, 2.
3. Divide the number by 2 as long as it is divisible.
4. Move to the next odd number after 2 (since even numbers greater than 2 are not prime).
5. Continue dividing and moving to the next integers until the square of the integer is greater than the dividend.
6. If any factor is greater than the square root of the original number and still divides the dividend, that factor is prime.

Python Example