Introduction:

Welcome, fellow coders! 🚀Ever wondered how the magic of prime numbers intertwines with computer science? You're in for a treat! 🎩✨In this blog post, we are tackling a classic problem: PRIME SUM. Buckle up as we dissect the efficient wizardry of the Sieve of Eratosthenes through a Java implementation. ready to dive into the code and uncover its secrets? Let's roll! 🤓💻 #PrimePairQuest #SieveSleuth

Understanding the Problem:

Given a positive integer A, our task is to find two prime numbers p1​ and p2​ such that p1 ​+ p2 ​= A. This problem is interesting because it combines concepts of prime number generation and basic arithmetic.

Sieve of Eratosthenes Overview:

The Sieve of Eratosthenes is a simple and efficient algorithm for finding all prime numbers up to a specified integer. It works by iteratively marking the multiples of each prime starting from 2, thereby sieving out the composite numbers and leaving behind the primes.

Step-by-Step Implementation:

Let's break down the Java implementation provided and understand each part:

public class Solution {
    public int[] primesum(int A) {
        // Step 1: Initialize an array to hold numbers same as their index from 0 to A 
        int[] eratosthenes = new int[A+1];
        for(int i=0; i<=A; i++){
            eratosthenes[i] = i;
        }

        // Step 2: Apply Sieve of Eratosthenes algorithm
        for(int i=2; i*i<=A; i++){
            if(eratosthenes[i]==i){
                for(int j=i*i; j<=A; j=j+i){
                    eratosthenes[j] = i;
                }
            }
        }

        // Step 3: Count the number of primes and store them in an array
        int prime = 0;
        for(int i=2; i<=A; i++){
            if(eratosthenes[i]==i){
                prime++;
            }
        }

        // Step 4: Create a new array of length of number of primes and store all the prime numbers in it
        int[] primeList = new int[prime];
        int idx = 0;
        for(int i=2; i<=A; i++){
            if(eratosthenes[i]==i){
                primeList[idx] = i;
                idx++;
            }
        }

        // Step 5: Create an array to store the smallest pair that sum up to A
        int[] ans = new int[2];

        // Step 6: Find prime pairs that sum up to A
        for(int i=0; i<prime; i++){
            for(int j=i; j<prime; j++){
                if((primeList[i]+primeList[j])==A){
                    ans[0] = primeList[i];
                    ans[1] = primeList[j];
                    break;  //Alternative "return ans;"
                }
            }
            if(ans[0]!=0){  //this condition breaks the outer loop as ans gets updated for the first time
                break;
            }
        }
        return ans;
    }
}

Explanation:

1. Array Initialization: We initialize an array eratosthenes with indices ranging from 0 to A, where each index initially holds its own value. This array will be used to mark numbers as we sieve them out.

2. Sieve of Eratosthenes: We iterate through the array starting from 2 up to the square root of A, marking multiples of each prime number as that prime number only. This step essentially identifies all primes up to A.

3. Count Primes and Store: We count the number of primes found and store them in an array named primeList.

4. Find Prime Pairs: We loop through the primeList array to find prime pairs that sum up to A. Once found, we store them in the ans array.

Conclusion:

The Sieve of Eratosthenes is a powerful algorithm for generating prime numbers efficiently. In this blog post, we've walked through a Java implementation of this algorithm to find prime pairs that sum up to a given number. Understanding such algorithms not only enhances problem-solving skills but also provides insights into the beauty of mathematics and computer science.

Happy coding!

Jordan ^_~