Generating a list of prime numbers up to a specified integer \( n \) in Python can be accomplished using various methods. Here, we’ll explore some of the most efficient algorithms, including the Sieve of Eratosthenes and a simple trial division method.
###### Method 1: Using the Sieve of Eratosthenes

The Sieve of Eratosthenes is an ancient algorithm that efficiently finds all prime numbers up to a specified integer. It works by iteratively marking the multiples of each prime number starting from 2.
Here is a Python implementation:
def sieve_of_eratosthenes(n):
primes = []
is_prime = [True] * (n + 1)
is_prime[0] = is_prime[1] = False # 0 and 1 are not prime numbers
for i in range(2, int(n**0.5) + 1):
if is_prime[i]:
for j in range(i * i, n + 1, i):
is_prime[j] = False
primes = [i for i in range(n + 1) if is_prime[i]]
return primes
# Example usage
n = 100
print(sieve_of_eratosthenes(n))

###### Method 2: Trial Division

For smaller values of \( n \), a simple trial division method can effectively generate prime numbers. This method checks each number for primality by testing divisibility with all integers up to its square root.
Here’s how this approach can be implemented in Python:
def is_prime(num):
if num < 2:
return False
for i in range(2, int(num ** 0.5) + 1):
if num % i == 0:
return False
return True
def primes_up_to_n(n):
primes = []
for num in range(2, n + 1):
if is_prime(num):
primes.append(num)
return primes
# Example usage
n = 100
print(primes_up_to_n(n))

###### Method 3: Using List Comprehension and *all()*

A more concise version of the trial division method can be constructed using list comprehension. This version leverages the *all()* function to check for primality.
def primes_up_to_n(n):
return [num for num in range(2, n + 1) if all(num % i != 0 for i in range(2, int(num ** 0.5) + 1))]
# Example usage
n = 100
print(primes_up_to_n(n))

###### Conclusion

The Sieve of Eratosthenes is the most efficient way for generating a list of primes for large values of \( n \), making it ideal for scenarios requiring all prime numbers up to a significant threshold. The trial division method, while simpler, is generally more manageable for smaller values of \( n \). Both methods effectively yield a list of prime numbers, allowing you to choose based on the specific requirements and constraints of your application.