def sieve_of_eratosthenes(limit):
    """
    Generate all prime numbers up to the given limit using the Sieve of Eratosthenes algorithm.
    """
    is_prime = [True] * (limit + 1)
    p = 2
    while (p * p <= limit):
        if (is_prime[p] == True):
            for i in range(p * p, limit + 1, p):
                is_prime[i] = False
        p += 1
    prime_numbers = [p for p in range(2, limit + 1) if is_prime[p]]
    return prime_numbers


def first_n_primes(n):
    """
    Calculate the first n prime numbers using the Sieve of Eratosthenes algorithm.
    """
    # Estimate an upper limit for the nth prime number using the approximation n * log(n * log(n))
    # This is a rough estimate and ensures we have a high enough limit to find the first n primes.
    import math
    if n < 6:
        limit = 15
    else:
        limit = int(n * math.log(n * math.log(n)))
    primes = sieve_of_eratosthenes(limit)
    return primes[:n]

# Example usage:
first_1000_primes = first_n_primes(1000)
print(first_1000_primes)