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

Here's a detailed explanation of the Java code:

Import Statements:

• import java.util.ArrayList;: Imports the ArrayList class for creating lists.
• import java.util.Collections;: Imports the Collections class for sorting lists.
• import java.util.List;: Imports the List interface for working with lists.

Class ZsigmondyNumbers:

1. main Method:

• Purpose: Main entry point of the program.
• Logic:
  • Creates a list of Pair objects representing different values for a and b in Zsigmondy numbers.
  • For each pair, prints the Zsigmondy numbers for n from 1 to 18.

2. zsigmondyNumber Method:

• Purpose: Calculates the Zsigmondy number for given values of n, a, and b.
• Logic:
  • Calculates dn = a^n - b^n.
  • If dn is prime, returns it.
  • Finds all divisors of dn.
  • Removes divisors that share a common factor with a^m - b^m for any m < n.
  • Returns the largest remaining divisor.

3. divisors Method:

• Purpose: Returns a list of all divisors of a given number.
• Logic:
  • Iterates from 1 up to the square root of the number.
  • If a divisor is found, adds it and its corresponding pair (if different) to the list.
  • Sorts the list and returns it.

4. isPrime Method:

• Purpose: Determines whether a given number is prime.
• Logic:
  • Handles basic cases for numbers less than or equal to 3.
  • Uses a 6k ± 1 optimization for primality testing.

5. gcd Method:

• Purpose: Calculates the greatest common divisor of two numbers using the Euclidean algorithm.
• Logic:
  • Repeatedly replaces the larger number with the remainder of their division until one becomes 0.
  • Returns the remaining non-zero number as the GCD.

Key Concepts:

• Zsigmondy Numbers: Numbers of the form a^n - b^n that have a prime divisor that doesn't divide a^m - b^m for any smaller m.
• Prime Number: A natural number greater than 1 that has no positive divisors other than 1 and itself.
• Greatest Common Divisor (GCD): The largest positive integer that divides two or more integers without a remainder.
• Modulo Operation (%): Used for checking the remainder after division.

import java.util.ArrayList;
import java.util.Collections;
import java.util.List;

public final class ZsigmondyNumbers {

	public static void main(String[] args) {
		record Pair(int a, int b) {}
		List<Pair> pairs = List.of( new Pair(2, 1), new Pair(3, 1), new Pair(4, 1), new Pair(5, 1),
			new Pair(6, 1), new Pair(7, 1), new Pair(3, 2), new Pair(5, 3), new Pair(7, 3), new Pair(7, 5) );	
		
		for ( Pair pair : pairs ) {
			System.out.println("Zsigmondy(n, " + pair.a + ", " + pair.b + ")");
		    for ( int n = 1; n <= 18; n++ ) {
		        System.out.print(zsigmondyNumber(n, pair.a, pair.b) + " ");
		    }
		    System.out.println(System.lineSeparator());
		}
		
	}
	
	private static long zsigmondyNumber(int n, int a, int b) {
		final long dn = (long) ( Math.pow(a, n) - Math.pow(b, n) );
		if ( isPrime(dn) ) {
			return dn;
		}
		
		List<Long> divisors = divisors(dn);
		for ( int m = 1; m < n; m++ ) {
		    long dm = (long) ( Math.pow(a, m) - Math.pow(b, m) );
		    divisors.removeIf( d -> gcd(dm, d) > 1 );
		}
		return divisors.get(divisors.size() - 1);
	}
	
	private static List<Long> divisors(long number) {
		List<Long> result = new ArrayList<Long>();
		for ( long d = 1; d * d <= number; d++ ) {
			if ( number % d == 0 ) {
			    result.add(d);
			    if ( d * d < number ) {
			    	result.add(number / d);
			    }
			}			        
		}
		Collections.sort(result);
		return result;
	}
	
	private static boolean isPrime(long number) {
		if ( number < 2 ) {
			return false;
		}
		if ( number % 2 == 0 ) {
			return number == 2;
		}
		if ( number % 3 == 0 ) {
			return number == 3;
		}
		
		int delta = 2;
		long k = 5;
		while ( k * k <= number ) {
		    if ( number % k == 0 ) {
		    	return false;
		    }
		    k += delta;
		    delta = 6 - delta;
		}
		return true;
	}
	
	private static long gcd(long a, long b) {
		while ( b != 0 ) {
			long temp = a; a = b; b = temp % b;
		}
		return a;
	}	

}