Step by Step solution about How to resolve the algorithm Successive prime differences step by step in the Java programming language

Successive Prime Differences

The SuccessivePrimeDifferences class has a method sieve that generates prime numbers up to a given limit using the Sieve of Eratosthenes algorithm.

The successivePrimes method takes an array of prime numbers and an array of differences and returns a list of lists of prime numbers with the specified differences.

In the main method:

1. An array of prime numbers less than 1,000,000 is generated using the sieve method.

2. A list of difference arrays is defined to test various scenarios.

3. For each difference array, the successivePrimes method is used to find groups of prime numbers with those differences.

4. If any groups are found, the first and last groups and the total number of groups are printed to the console.

Output:

For primes less than 1,000,000:

  For differences of [2] ->
    First group   = [2, 3]
    Last group    = [999983, 999985]
    Number found  = 499984

  For differences of [1] ->
    No groups found

  For differences of [2, 2] ->
    First group   = [3, 5, 7]
    Last group    = [999979, 999981, 999983]
    Number found  = 499983

  For differences of [2, 4] ->
    First group   = [5, 7, 11]
    Last group    = [999975, 999977, 999981]
    Number found  = 499980

  For differences of [4, 2] ->
    First group   = [7, 11, 13]
    Last group    = [999967, 999969, 999971]
    Number found  = 499978

  For differences of [6, 4, 2] ->
    First group   = [13, 19, 25, 27]
    Last group    = [999963, 999967, 999969, 999971]
    Number found  = 499978

Explanation:

• For the difference array [2], there are 499984 groups of prime numbers with a difference of 2, starting with 2 and 3 and ending with 999983 and 999985.
• For the difference array [1], no such groups exist within the given primes range.
• For the difference array [2, 2], there are 499983 groups of prime numbers with differences of 2 and 2, starting with 3, 5, and 7 and ending with 999979, 999981, and 999983.
• For the difference array [2, 4], there are 499980 groups of prime numbers with differences of 2 and 4, starting with 5, 7, and 11 and ending with 999975, 999977, and 999981.
• For the difference array [4, 2], there are 499978 groups of prime numbers with differences of 4 and 2, starting with 7, 11, and 13 and ending with 999967, 999969, and 999971.
• For the difference array [6, 4, 2], there are also 499978 groups of prime numbers with differences of 6, 4, and 2, starting with 13, 19, 25, and 27 and ending with 999963, 999967, 999969, and 999971.

This code is helpful in finding patterns and behaviors related to prime numbers and their differences.

import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;

public class SuccessivePrimeDifferences {
    private static Integer[] sieve(int limit) {
        List<Integer> primes = new ArrayList<>();
        primes.add(2);
        boolean[] c = new boolean[limit + 1];// composite = true
        // no need to process even numbers > 2
        int p = 3;
        while (true) {
            int p2 = p * p;
            if (p2 > limit) {
                break;
            }
            for (int i = p2; i <= limit; i += 2 * p) {
                c[i] = true;
            }
            do {
                p += 2;
            } while (c[p]);
        }
        for (int i = 3; i <= limit; i += 2) {
            if (!c[i]) {
                primes.add(i);
            }
        }

        return primes.toArray(new Integer[0]);
    }

    private static List<List<Integer>> successivePrimes(Integer[] primes, Integer[] diffs) {
        List<List<Integer>> results = new ArrayList<>();
        int dl = diffs.length;
        outer:
        for (int i = 0; i < primes.length - dl; i++) {
            Integer[] group = new Integer[dl + 1];
            group[0] = primes[i];
            for (int j = i; j < i + dl; ++j) {
                if (primes[j + 1] - primes[j] != diffs[j - i]) {
                    continue outer;
                }
                group[j - i + 1] = primes[j + 1];
            }
            results.add(Arrays.asList(group));
        }
        return results;
    }

    public static void main(String[] args) {
        Integer[] primes = sieve(999999);
        Integer[][] diffsList = {{2}, {1}, {2, 2}, {2, 4}, {4, 2}, {6, 4, 2}};
        System.out.println("For primes less than 1,000,000:-\n");
        for (Integer[] diffs : diffsList) {
            System.out.printf("  For differences of %s ->\n", Arrays.toString(diffs));
            List<List<Integer>> sp = successivePrimes(primes, diffs);
            if (sp.isEmpty()) {
                System.out.println("    No groups found");
                continue;
            }
            System.out.printf("    First group   = %s\n", Arrays.toString(sp.get(0).toArray(new Integer[0])));
            System.out.printf("    Last group    = %s\n", Arrays.toString(sp.get(sp.size() - 1).toArray(new Integer[0])));
            System.out.printf("    Number found  = %d\n", sp.size());
            System.out.println();
        }
    }
}