**π‘ Problem Formulation:** When working with number theory or coding algorithms in Python, a common problem is to find the number of prime numbers within a given range. Assume you are given the range from 10 to 50 and you want to count how many prime numbers exist between these two bounds. The desired output would be a single integer representing this count.

## Method 1: The Classic Iterative Approach

This method involves checking each number in the given range to see if it is prime. A number is considered prime if it has exactly two distinct positive divisors: 1 and itself. The process includes a nested loop where the inner loop checks divisibility by all integers up to the square root of the number being tested.

Here’s an example:

def is_prime(n): if n <= 1: return False for i in range(2, int(n**0.5)+1): if n % i == 0: return False return True def count_primes(start, end): return sum(1 for i in range(start, end+1) if is_prime(i)) # Example usage: print(count_primes(10, 50))

Output: 11

This code snippet defines a function `is_prime()`

that determines if a number is prime. It then defines a function `count_primes()`

that counts how many primes exist within a given range. It uses list comprehension to tally up the prime counts between 10 and 50, yielding the output 11.

## Method 2: Sieve of Eratosthenes

The Sieve of Eratosthenes is an efficient algorithm to find all prime numbers up to any given limit. It works by iteratively marking the multiples of each prime number starting from 2. The remaining unmarked numbers are primes, and this method is excellent for handling large ranges efficiently.

Here’s an example:

def sieve_of_eratosthenes(end): prime = [True for _ in range(end+1)] p = 2 while p * p <= end: if prime[p]: for i in range(p * p, end+1, p): prime[i] = False p += 1 return [p for p in range(2, end) if prime[p]] # Example usage: print(len(sieve_of_eratosthenes(50)) - len(sieve_of_eratosthenes(10)))

Output: 11

Here, a function `sieve_of_eratosthenes()`

constructs a list of prime numbers up to a specified end point. By taking the length of primes up to 50 and subtracting the length of primes up to just before 10, we find the number of primes between 10 and 50, which is 11 in this case.

## Method 3: Using Functions From the `sympy`

Library

The `sympy`

library comes with built-in functions for number theory, including `primerange()`

, which generates all prime numbers in the given range. This method is simple, fast, and requires significantly less code due to the utility of the `sympy`

library.

Here’s an example:

from sympy import primerange # Example usage: primes = list(primerange(10, 50)) print(len(primes))

Output: 11

In this snippet, the `primerange()`

function from the `sympy`

library is used to generate a list of prime numbers between 10 and 50. The length of this list is then printed, revealing there are 11 primes in this range.

## Method 4: Using Itertools and a Custom Prime Checker

This combines Python’s `itertools`

library with a custom-written prime-checking function. The `itertools`

library is used to iterate over the range efficiently, while the custom prime checker verifies each number. This method strikes a balance between code readability and performance.

Here’s an example:

from itertools import compress, count def is_prime(n): return n > 1 and all(n % i for i in range(2, int(n**0.5) + 1)) def count_primes(start, end): return sum(1 for i in compress(count(start), map(is_prime, range(start, end+1)))) # Example usage: print(count_primes(10, 50))

Output: 11

The function `is_prime()`

returns True if a number is prime. The `count_primes()`

function then leverages `compress()`

and `count()`

from the `itertools`

library to iterate through the range, applying the prime checker to each number and tallying the count of primes found.

## Bonus One-Liner Method 5: Functional Approach with Filter and Lambda

This one-liner method uses the power of Python’s functional programming features. It applies a `lambda`

function to filter out non-prime numbers within the range, resulting in a list of primes. This concise method is elegant but may not be the most efficient solution for large ranges.

Here’s an example:

print(len(list(filter(lambda x: all(x % i != 0 for i in range(2, int(x**0.5) + 1)), range(10, 51)))))

Output: 11

The code uses `filter()`

with a lambda function that checks if a number is prime. The range is processed to leave only prime numbers, which are then counted to give the final output. This one-liner is the compact version of the iterative methods described in earlier sections.

## Summary/Discussion

**Method 1:**The Classic Iterative Approach. Strengths: Straightforward and easy to understand. Weaknesses: Inefficient for large ranges as it checks each number individually.**Method 2:**Sieve of Eratosthenes. Strengths: Highly efficient, particularly for larger ranges. Weaknesses: Requires more memory as it stores a list for marking non-primes.**Method 3:**Using Functions From the`sympy`

Library. Strengths: Extremely concise and easy to implement. Weaknesses: Requires an external library, which may not be desirable for minimal dependency solutions.**Method 4:**Using Itertools and a Custom Prime Checker. Strengths: Readable and makes good use of built-in libraries for better performance. Weaknesses: Slightly more complex due to the combination of tools and custom code.**Method 5:**Functional Approach with Filter and Lambda. Strengths: Elegant and concise. Weaknesses: Potentially inefficient for very large ranges and not as readable as more verbose methods.