Step by Step solution about How to resolve the algorithm Fortunate numbers step by step in the Java programming language

The Java program is designed to identify and display the first 50 fortunate numbers. Fortunate numbers are integers that, when added to the primorial of the preceding prime number, result in a probable prime. The program generates and stores the fortunate numbers in a NavigableSet.

1. Prime Sieve (primeSieve** method):**

   • Utilizes a BitSet to represent a sieve of prime numbers up to a specified number (aNumber).
   • Begins with all bits set to true, indicating potential primes.
   • Iterates through potential primes from 2 to the square root of aNumber.
   • For each prime found, it marks its multiples as false in the sieve, eliminating them as potential primes.
   • Returns the resulting BitSet with prime numbers marked as true.

2. Fortunate Number Generation:

   • The main method initializes a NavigableSet (fortunates) for storing fortunate numbers.
   • The program initializes a BigInteger variable primorial to 1, representing the product of all prime numbers so far.
   • In a loop, it iterates through prime numbers:
     • Calculates a new primorial by multiplying it with the current prime.
     • Finds the next odd integer (candidate) by adding 2 and keeps incrementing it until adding it to primorial results in a probable prime.
     • Adds the candidate to the fortunates set.

3. Output:

   • In the main method, the program prints the first 50 distinct fortunate numbers to the console, formatted with a width of 4 characters and a newline every 10th number.

4. Additional Constants and Methods:

   • CERTAINTY_LEVEL: Used as a parameter for BigInteger.isProbablePrime to define the confidence level in finding probable primes.

import java.math.BigInteger;
import java.util.BitSet;
import java.util.NavigableSet;
import java.util.TreeSet;

public final class FortunateNumbers {

	public static void main(String[] aArgs) {
		BitSet primes = primeSieve(400);		
		NavigableSet<Integer> fortunates = new TreeSet<Integer>();
        BigInteger primorial = BigInteger.ONE;
		
		for ( int prime = 2; prime >= 0; prime = primes.nextSetBit(prime + 1) ) {
		    primorial = primorial.multiply(BigInteger.valueOf(prime));
		    int candidate = 3;
		    while ( ! primorial.add(BigInteger.valueOf(candidate)).isProbablePrime(CERTAINTY_LEVEL) ) {
		    	candidate += 2;
		    }
		    fortunates.add(candidate);
		}
		
		System.out.println("The first 50 distinct fortunate numbers are:");
		for ( int i = 0; i < 50; i++ ) {
			System.out.print(String.format("%4d%s", fortunates.pollFirst(), ( i % 10 == 9 ? "\n" : "" )));
		}
	}
	
	private static BitSet primeSieve(int aNumber) {
		BitSet sieve = new BitSet(aNumber + 1);
		sieve.set(2, aNumber + 1);
		for ( int i = 2; i <= Math.sqrt(aNumber); i = sieve.nextSetBit(i + 1) ) {
			for ( int j = i * i; j <= aNumber; j = j + i ) {
				sieve.clear(j);
			}
		}		
		return sieve;
	}
	
	private static final int CERTAINTY_LEVEL = 10;

}