Question

\( (\text { Sieve of Eratosthenes) } \text { A prime number is any integer greater than } 1\) that is evenly divisible only by itself and \(1 .\) The Sieve of Eratosthenes is a method of finding prime numbers. It operates as follows: a) Create a primitive type boolean array with all elements initialized to true. Array elements with prime indices will remain true. All other array elements will eventually be set to false. b) Starting with array index \(2,\) determine whether a given element is true. If so, loop through the remainder of the array and set to false every element whose index is a multiple of the index for the element with value true. Then continue the process with the next element with value true. For array index \(2,\) all elements beyond element 2 in the array that have indices which are multiples of 2 (indices \(4,6,8,10,\) etc.) will be set to false; for array index 3 , all elements beyond element 3 in the array that have indices which are multiples of 3 (indices 6,9,12,15 , etc.) will be set to \(f\) alse; and so on. When this process completes, the array elements that are still true indicate that the index is a prime number. These indices can be displayed. Write an application that uses an array of 1000 elements to determine and display the prime numbers between 2 and \(999 .\) Ignore array elements 0 and 1.

Short answer

Use a boolean array to find and display prime numbers from 2 to 999.

Step-by-step solution

Initialize the Boolean Array

Create a boolean array `isPrime` with 1000 elements and initialize all elements to `true`. This array will keep track of indices corresponding to prime numbers. Indices 0 and 1 should be set to `false` immediately since neither 0 nor 1 is prime.

Start Sieve from First Prime

Begin the sieve process from index 2, which is the smallest prime number. Check if the current index (starting at 2) is set to `true` in the `isPrime` array.

Mark Multiples of Current Number

If the `isPrime[index]` is `true`, iterate through the array beyond this index and set to `false` all indices that are multiples of the current index. For example, starting at index 2, set indices 4, 6, 8, up to 999 that are multiples of 2 to `false`.

Iterate and Repeat

Move to the next index and repeat the process of checking if the index is `true` in the `isPrime` array and mark its multiples as `false`. Continue this until you reach the end of the array.

Identify Prime Numbers

After completing the sieve process, iterate through the `isPrime` array starting from index 2 to 999. Collect the indices that remain `true` as these are the prime numbers.

Display Prime Numbers

Output or display the prime numbers identified from the `isPrime` array. These are the indices that have remained `true`, indicating they are prime numbers.

Key concepts

Prime Numbers

Prime numbers are fascinating divisors in mathematics. They are numbers greater than 1 that have no divisors other than 1 and themselves. This uniqueness sets them apart from composite numbers, which have additional divisors. Prime numbers are foundational for various mathematical concepts and have numerous applications in fields such as cryptography.
They are often considered the building blocks of natural numbers, similar to how atoms are the building blocks of matter. Identifying prime numbers is a common task in number theory and helps in solving many mathematical problems. Examples of prime numbers include 2, 3, 5, 7, and 11. One distinct feature is that 2 is the only even prime number, while others are odd.
Detecting prime numbers, especially large ones, requires efficient algorithms since manual identification isn't practical. This is where methods like the Sieve of Eratosthenes come into play, providing systematic approaches for prime determination.

Boolean Array

A boolean array is a collection of 'true' or 'false' values. In computer programming, boolean arrays are useful for tracking binary states—something is true, or it isn't. When implementing the Sieve of Eratosthenes, a boolean array plays a critical role.
The array begins with each element set to `true`, representing potential primality for each index. Indices 0 and 1 are immediately set to `false` since they are not prime numbers. This boolean setup simplifies the process of identifying primes. By flipping elements to `false`, the algorithm efficiently marks non-prime numbers.
Each index in a boolean array corresponds to a number, where the index itself suggests its primality status. This setup makes sieve algorithms fast and minimal in storage requirement, only necessitating a single type of data check: true or false.

Prime Identification

Prime identification is at the heart of understanding which numbers are prime using algorithms like the Sieve of Eratosthenes. This method centers on determining whether a number can be divided evenly by any number other than 1 and itself. To achieve this efficiently, the algorithm leverages the boolean array. Using the sieve, we repeatedly "sweep" through an array of indices corresponding to numbers, each time marking multiples of a confirmed prime as non-prime. By the end of the sieve process, the indices that remain set to `true` directly correlate to prime numbers.
The algorithm's beauty lies in its ability to limit divisibility checks—by the time it processes each prime, multiples up to the maximum limit are already marked, obviating redundant checks. This approach sacrifices space (for the boolean array) for an increase in speed, making it a powerful method for prime identification in large ranges.

Algorithm Design

Algorithm design is a fundamental part of creating effective solutions in computer science. The Sieve of Eratosthenes is a classic example of excellent algorithm design. It elegantly balances efficiency and simplicity, making it a benchmark for teaching algorithm efficiency and complexity.
In designing this algorithm, the primary goal is to reduce the overhead of repeated calculations. This is achieved by initializing a large boolean array to track potential primes and iterating over this array to systematically mark non-primes. Unlike simple trial division, the sieve design minimizes the operations needed to identify prime numbers in a given range.
The iterative marking of non-prime multiples is cautious of time complexity, providing a relatively low-cost solution compared to naive primality testing. This makes the sieve an ideal algorithm for learning basic design strategies, especially those that emphasize space-time trade-offs and systematic problem solving.