How Do You Calculate Prime Numbers

Sieve of Eratosthenes Algorithm:

\[ \text{For } n > 1, \text{ find all primes } \leq n \text{ by iteratively marking multiples} \]

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 is one of the most efficient ways to find small primes and was created by the Greek mathematician Eratosthenes in the 3rd century BC.

2. How Does the Algorithm Work?

The algorithm works by iteratively marking the multiples of each prime number starting from 2:

\[ \text{For } i = 2 \text{ to } \sqrt{n}, \text{ mark all multiples of } i \text{ as composite} \]

Key Steps:

Create a list of consecutive integers from 2 to n
Start with the first number (2) - it's prime
Mark all multiples of 2 as composite
Move to the next unmarked number (3) - it's prime
Mark all multiples of 3 as composite
Repeat until you reach √n
All remaining unmarked numbers are prime

3. Algorithm Steps

Detailed Process:

1. Create a boolean array of size n+1, initialize all as true
2. Mark positions 0 and 1 as false (not prime)
3. For i from 2 to √n:
- If isPrime[i] is true, mark all multiples of i as false
4. All remaining true positions are prime numbers

4. Using the Calculator

Instructions: Enter an upper limit (n) between 2 and 10,000. The calculator will find all prime numbers up to that limit using the Sieve of Eratosthenes algorithm and display the results with execution time.

5. Frequently Asked Questions (FAQ)

Q1: What is the time complexity of this algorithm?
A: The Sieve of Eratosthenes has a time complexity of O(n log log n), making it very efficient for finding primes up to moderate limits.

Q2: What is the space complexity?
A: The space complexity is O(n) as it requires an array of size n+1 to track prime status.

Q3: Why do we only check up to √n?
A: Any composite number n must have a prime factor less than or equal to √n, so checking beyond that is unnecessary.

Q4: What are the limitations of this algorithm?
A: The main limitation is memory usage for very large n. For extremely large numbers, other algorithms like Miller-Rabin primality test are more suitable.

Q5: Can this algorithm be optimized?
A: Yes, optimizations include checking only odd numbers after 2, and starting marking from i² rather than 2*i.