How does the efficiency of different prime factorization methods compare in Python?

Python: Efficient Prime Factorization

Question 1:
Understanding an existing Python program that calculates the largest prime factor of 600851475143, and exploring alternative prime factorization methods.

Answer:
The code you found online operates efficiently by repeatedly dividing the number by its smallest prime factor until it reaches the largest prime factor. While the number is not divisible by the current prime factor, it continues to increment the prime factor.

An alternate method is to use a brute-force approach:

<code class="python">def prime_factors(n):
    i = 2
    factors = []
    while i * i <= n:
        if n % i:
            i += 1
        else:
            n //= i
            factors.append(i)
    if n > 1:
        factors.append(n)
    return factors</code>

This function tests every number from 2 to the square root of the given number to determine its prime factors. However, this method is less efficient for large numbers.

Question 2:
Comparing the efficiency of the two provided code snippets.

Answer:
The second code snippet, which simply increments a counter, is much slower because it checks every integer up to a certain value, whereas the first code snippet checks only the smallest prime factor and immediately divides by it, efficiently eliminating that factor.

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