How To Calculate Prime Numbers In Java

Sieve of Eratosthenes Algorithm:

\[ \text{Mark multiples of primes up to } \sqrt{n} \text{ to find all primes up to n} \]

Total Prime Numbers Found:

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1. What Is The Sieve Of Eratosthenes?

The Sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit. It works by iteratively marking the multiples of each prime number starting from 2, leaving only the prime numbers unmarked.

2. How Does The Algorithm Work?

The algorithm follows these steps:

\[ \text{1. Create list of integers from 2 to n} \] \[ \text{2. Start with first prime number p = 2} \] \[ \text{3. Mark all multiples of p as composite} \] \[ \text{4. Find next unmarked number > p, set as new p} \] \[ \text{5. Repeat until } p^2 > n \]

Time Complexity: O(n log log n)
Space Complexity: O(n)

3. Java Implementation

Java Code Example:

public static List sieveOfEratosthenes(int n) {
    boolean[] isPrime = new boolean[n + 1];
    Arrays.fill(isPrime, true);
    isPrime[0] = isPrime[1] = false;
    
    for (int i = 2; i * i <= n; i++) {
        if (isPrime[i]) {
            for (int j = i * i; j <= n; j += i) {
                isPrime[j] = false;
            }
        }
    }
    
    List primes = new ArrayList<>();
    for (int i = 2; i <= n; i++) {
        if (isPrime[i]) {
            primes.add(i);
        }
    }
    return primes;
}

4. Using The Calculator

Instructions: Enter an integer n (2 ≤ n ≤ 10000) to find all prime numbers up to n using the Sieve of Eratosthenes algorithm.

5. Frequently Asked Questions (FAQ)

Q1: Why is the Sieve of Eratosthenes efficient?
A: It eliminates multiples of each prime only once, achieving O(n log log n) time complexity, making it very efficient for finding primes up to large numbers.

Q2: What is the maximum limit for this calculator?
A: The calculator is limited to n = 10000 for performance reasons, but the algorithm can handle much larger numbers.

Q3: How does it compare to trial division?
A: Sieve of Eratosthenes is much faster for finding all primes up to n, while trial division is better for checking if a single number is prime.

Q4: Can this algorithm be optimized further?
A: Yes, optimizations include skipping even numbers, using bit arrays for memory efficiency, and segmented sieve for very large n.

Q5: What are some practical applications?
A: Cryptography, number theory research, mathematical computations, and algorithms that require prime number generation.