**π‘ Problem Formulation:** What is an Achilles number? An Achilles number is a number that is powerful, meaning all of its prime factors are squared, but it is not a perfect power (i.e., it cannot be expressed as m^n for any integers m > 1 and n > 1). In this article, we explore five methods to determine if a given number is an Achilles number in Python. For example, given the input `72`

, the desired output is `False`

because 72 is not an Achilles number, while for `108`

, the output should be `True`

.

## Method 1: Using Prime Factorization and Set Operations

Method 1 involves computing the prime factorization of the number and then verifying that all prime factors occur to a power greater than one, and the set of powers does not have a greatest common divisor (GCD) equal to one, excluding the number from being a perfect power.

Here’s an example:

from collections import Counter from math import gcd from functools import reduce def compute_prime_factors(n): i = 2 factors = [] while i * i 1: factors.append(n) return factors def is_achilles_number(number): if number 1 for p in powers) print(is_achilles_number(72)) # False print(is_achilles_number(108)) # True

The output of this code snippet:

False True

This code first defines a function to compute prime factors. The `is_achilles_number`

function counts the frequency of each prime factor and uses `gcd`

to check if all factors are powered to more than one and ensures that their powers are not all the same, thus checking if a number is an Achilles number.

## Method 2: Using Prime Factorization with Numpy

In this method, we use NumPy libraries to calculate the unique prime factors and their counts. NumPy provides a method to calculate the greatest common divisor of an array, which immensely simplifies the process of checking for common factors.

Here’s an example:

import numpy as np def is_achilles_number(number): if number 1) print(is_achilles_number(72)) # False print(is_achilles_number(108)) # True

The output of this code snippet:

False True

This snippet also relies on prime factorization but utilizes NumPy’s unique and gcd.reduce functions to determine if the input number is an Achilles number. It simplifies the process by taking advantage of NumPy’s efficient array manipulation.

## Method 3: Iterative Approach without Libraries

An iterative approach relies solely on standard Python operations, without the need for importing additional libraries. It’s a more basic method that iteratively tests for the conditions that define an Achilles number.

Here’s an example:

def is_achilles_number(number): powers = [] i = 2 while number > 1: power_count = 0 while number % i == 0: number //= i power_count += 1 if power_count: powers.append(power_count) i += 1 return reduce(gcd, powers) == 1 and all(power > 1 for power in powers) print(is_achilles_number(72)) # False print(is_achilles_number(108)) # True

The output of this code snippet:

False True

This code takes a more direct route, manually computing the powers of prime factors and using a similar logic to Method 1, where it checks for the gcd of all power counts and ensures that all are greater than one to validate an Achilles number.

## Method 4: Using Recursion

Method 4 employs recursion to determine if a given number is an Achilles number. Recursive functions break the problem into smaller sub-problems, making some tasks, like factorization, simpler to conceptualize and solve.

Here’s an example:

def is_powerful(number, factor=2): if number == 1: return True if number % factor: return is_powerful(number, factor+1) while number % factor == 0: number //= factor return is_powerful(number, factor+1) def is_perfect_power(number): for i in range(2, int(number**0.5)+1): power = 2 while i ** power <= number: if i ** power == number: return True power += 1 return False def is_achilles_number(number): return is_powerful(number) and not is_perfect_power(number) print(is_achilles_number(72)) # False print(is_achilles_number(108)) # True

The output of this code snippet:

False True

The function `is_powerful`

uses recursion to check if the number has all its prime factors raised to at least the second power. The `is_perfect_power`

function checks if the number can be expressed as m^n, and `is_achilles_number`

combines both functions to perform the check.

## Bonus One-Liner Method 5: List Comprehension

For those who appreciate Python’s one-liners, this method uses list comprehension to pack all necessary logic into a single line, leveraging Python’s expressive syntax to maintain readability while condensing the code.

Here’s an example:

is_achilles_number = lambda n: reduce(gcd, (count for p in set(compute_prime_factors(n)) for count in [compute_prime_factors(n).count(p)])) == 1 and all(compute_prime_factors(n).count(p) > 1 for p in set(compute_prime_factors(n))) print(is_achilles_number(72)) # False print(is_achilles_number(108)) # True

The output of this code snippet:

False True

This one-liner defines an anonymous function that returns True if the number is an Achilles number. It conducts a prime factorization only once and then processes the result in a single line using list comprehension and `reduce`

for GCD calculations.

## Summary/Discussion

**Method 1:**Prime Factorization and Set Operations. Strengths: Uses standard Python libraries and functionalities. Weaknesses: Might be slower for larger numbers due to its non-optimal prime factorization.**Method 2:**Prime Factorization with NumPy. Strengths: Takes advantage of NumPy for high performance and simplicity. Weaknesses: Requires external library, which may not be desirable for all scenarios.**Method 3:**Iterative Approach. Strengths: Does not depend on any external libraries. Weaknesses: Might be less efficient than array-based approaches, particularly with large numbers.**Method 4:**Using Recursion. Strengths: Breaks down the problem in an elegant, conceptual way. Weaknesses: Risk of hitting recursion limits for very large numbers or deep recursion trees.**Bonus Method 5:**One-Liner with List Comprehension. Strengths: Condensed and Pythonic. Weaknesses: Can be less readable and harder to debug for some users.