**π‘ Problem Formulation:** The task at hand is to develop a Python program that is capable of generating a sequence of five random prime numbers within the range of 100 to 150. The desired output is a series where each element is a unique prime number such as [101, 107, 113, 131, 149].

## Method 1: Brute Force Approach

Using a brute force approach, we can iterate through numbers between 100 to 150, check each for primality, and accumulate a list of prime numbers. Then, we randomly select five from this list. This method relies on checking factors of each candidate number to determine if it is prime.

Here’s an example:

import random 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 generate_primes(start, end, num): primes = [i for i in range(start, end + 1) if is_prime(i)] return random.sample(primes, num) five_primes = generate_primes(100, 150, 5)

Output: [103, 107, 113, 137, 149]

This code snippet defines two functions: `is_prime()`

to check primality and `generate_primes()`

to generate the prime series. The `generate_primes()`

function creates a list of prime numbers between the given `start`

and `end`

and then uses `random.sample()`

to pick five unique primes.

## Method 2: Sieve of Eratosthenes

Implementing the Sieve of Eratosthenes algorithm, which systematically eliminates the non-prime numbers, is a time-efficient way to find all primes within a given range. From this list, five primes are randomly chosen.

Here’s an example:

import random def sieve_of_eratosthenes(end): prime = [True for i in range(end+1)] p = 2 while (p * p <= end): if (prime[p] == True): for i in range(p * p, end+1, p): prime[i] = False p += 1 return [p for p in range(100, end) if prime[p]] random_primes = random.sample(sieve_of_eratosthenes(150), 5)

Output: [101, 103, 109, 127, 139]

The `sieve_of_eratosthenes()`

function generates a list of booleans where the index is marked `False`

if it is not a prime and `True`

if it is a prime. We then create a list of prime numbers from 100 to 150 and select five random primes from it.

## Method 3: Increment and Check

By incrementing from the lowest possible prime within the range, checking each for primality, and stopping when enough primes are found, this method ensures fewer calculations than a brute force across the entire range. This is a hybrid between brute force and optimization.

Here’s an example:

import random def is_prime(num): return all(num % i != 0 for i in range(2, int(num ** 0.5) + 1)) primes = [] number = 101 while len(primes) < 5: if is_prime(number): primes.append(number) number += 2 # increment by 2, since even numbers are not prime random_primes = primes

Output: [101, 103, 107, 109, 113]

The `is_prime()`

function returns `True`

if a number is prime, which is confirmed by trying to divide by all numbers up to its square root. By incrementing `number`

by 2, we ensure only odd numbers are checked (as even numbers cannot be prime), which reduces computational load.

## Method 4: Using itertools and Optimized Prime Check

Utilizing Python’s `itertools`

module, we can create efficient loops for generating primes. The `islice`

function ensures we inspect only the numbers within the specified range, and an optimized prime check reduces the number of divisions we need to perform.

Here’s an example:

import random from itertools import islice, count def is_prime(n): return n > 1 and all(n % i for i in islice(count(2), int(n**0.5)-1)) primes = [n for n in islice(count(100), 50) if is_prime(n)] five_primes = random.sample(primes, 5)

Output: [107, 109, 113, 131, 149]

This code snippet uses `itertools`

to efficiently create an iterator that starts counting from 100. For each number, the `is_prime()`

function determines if it is prime by checking if it has no divisors up to its square root. Finally, five primes are chosen at random from the list of primes.

## Bonus One-Liner Method 5: Comprehension and Randomness

A compact one-liner method combines list comprehension with `random.sample`

to filter and select primes with minimal code.

Here’s an example:

import random five_primes = random.sample([p for p in range(100, 151) if all(p % d != 0 for d in range(2, int(p**0.5) + 1))], 5)

Output: [101, 103, 107, 127, 139]

This example utilizes a list comprehension inside the `random.sample()`

function to check each number between 100 and 150 for primality before randomly selecting five primes. It’s an elegant solution but may not be as readable for those unfamiliar with comprehensions.

## Summary/Discussion

**Method 1:**Brute Force Approach. Straightforward but potentially inefficient for larger ranges. Good for small datasets.**Method 2:**Sieve of Eratosthenes. Extremely efficient for generating many primes. Overkill for this particular task but great for educational purposes.**Method 3:**Increment and Check. Balances between efficiency and simplicity. Less work than brute force but can be slow for large datasets.**Method 4:**Using itertools and Optimized Prime Check. Offers high efficiency with advanced Python features. Best for those comfortable with`itertools`

and generator expressions.**Bonus One-Liner Method 5:**Comprehension and Randomness. Quick and concise. Ideal for compact code but sacrifices some readability and may be slower than other methods.

Emily Rosemary Collins is a tech enthusiast with a strong background in computer science, always staying up-to-date with the latest trends and innovations. Apart from her love for technology, Emily enjoys exploring the great outdoors, participating in local community events, and dedicating her free time to painting and photography. Her interests and passion for personal growth make her an engaging conversationalist and a reliable source of knowledge in the ever-evolving world of technology.