How Can We Efficiently Generate Prime Numbers?

Generate Prime Numbers with Elegance and Efficiency

In the realm of programming, finding an elegant and efficient way to generate prime numbers is a classic challenge. Let's explore an approach that strikes a balance between conciseness and performance.

Consider using the prime number theorem, which estimates the number of primes less than or equal to n as pi(n) ≈ n / log(n). This estimate provides an upper bound on the size of a sieve that can be used to identify the primes.

The sieve method, also known as the Sieve of Eratosthenes, iterates through a range of numbers and eliminates all non-primes by marking them as composite. For this task, we can utilize a BitSet to represent the set of primes, with each bit corresponding to a number in the range.

Below is a Java implementation of this elegant and efficient prime number generation method:

<code class="java">public static BitSet computePrimes(int limit) {
    BitSet primes = new BitSet();
    primes.set(0, false);
    primes.set(1, false);
    primes.set(2, limit, true);
    for (int i = 0; i * i < limit; i++) {
        if (primes.get(i)) {
            for (int j = i * i; j < limit; j += i) {
                primes.clear(j);
            }
        }
    }
    return primes;
}</code>

This method efficiently generates the first million primes in approximately a second on a typical laptop. Its combination of precision and speed makes it a valuable tool for generating prime numbers in various computing scenarios.

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