The Sieve of Eratosthenes is an ancient algorithm used to find all prime numbers up to a specified integer. It works by iteratively marking the multiples of each prime number starting from 2. The numbers that remain unmarked at the end of the process are the prime numbers.Steps of the Sieve of Eratosthenes:Create a list of consecutive integers from 2 to the desired limit ( n ).Start with the first number in the list (2) and mark all of its multiples.Move to the next unmarked number and repeat the marking process for its multiples.Continue this until you have processed numbers up to the square root of ( n ).The remaining unmarked numbers in the list are the prime numbers.Greek MathematicianThe method is named after the ancient Greek mathematician Eratosthenes (c. 276–194 BC), who is credited with its invention.

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Sieve of Eratosthenes

The Sieve of Eratosthenes is a simple and efficient algorithm used to find all prime numbers up to a given limit. It is named after the ancient Greek mathematician Eratosthenes. This algorithm is one of the most popular methods for finding primes and is still used today in various applications.

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Algorithm Explanation

- Step 1: Create a list of consecutive integers from 2 through the limit.
- Step 2: Start with the first prime number, 2, and mark all of its multiples as composite.
- Step 3: Move to the next unmarked number (which is a prime) and repeat the process.
- Step 4: Continue this process until all numbers up to the limit have been processed.

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Example

Let's find all prime numbers up to 30 using the Sieve of Eratosthenes:
- Start with 2 as the first prime number and mark all its multiples: 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30.
- Move to the next unmarked number, which is 3, and mark all its multiples: 9, 15, 21, 27.
- Repeat this process for the remaining unmarked numbers: 5, 7, 11, 13, 17, 19, 23, 29.

The remaining unmarked numbers are all prime: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

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Conclusion

The Sieve of Eratosthenes is a straightforward and efficient method for finding prime numbers within a given range. It eliminates the need for repeated division operations and is a great tool for generating prime numbers quickly. This algorithm has stood the test of time and is still widely used in modern mathematics and computer science.