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

The provided Java code is designed to explore Curzon numbers, which are positive integers satisfying a specific mathematical criterion. The program calculates and displays both generalized Curzon numbers and the 1,000th generalized Curzon number for various bases (k) ranging from 2 to 10 in increments of 2.

Here's a detailed breakdown of the code:

1. main Method:

   • The main method serves as the program's entry point. It begins by iterating over a range of base values (k) from 2 to 10 in steps of 2.

2. Output for Each Base:

   • For each base, the program prints a header indicating the base being used.
   • It initializes variables n (the current number being checked) and count (the number of Curzon numbers found so far) to 1 and 0, respectively.
   • A while loop continues until count reaches 50. Inside the loop, it checks if the current number (n) is a generalized Curzon number with the given base (k). If so, it prints n and increments count. The output is formatted to show 10 numbers per line.

3. Calculating the 1,000th Generalized Curzon Number:

   • After finding the first 50 generalized Curzon numbers, the program continues the loop until count reaches 1,000. This loop increments n and count only when a generalized Curzon number is found.
   • Finally, it prints the 1,000th generalized Curzon number found for the current base.

4. isGeneralisedCurzonNumber Method:

   • This method checks if a given number (aN) is a generalized Curzon number with base aK. It calculates r as aK * aN and then uses the modulusPower method to compute aK^aN % (r + 1). If the result is equal to r, aN is a generalized Curzon number, and the method returns true. Otherwise, it returns false.

5. modulusPower Method:

   • The modulusPower method computes the modular exponentiation of a base (aBase) raised to an exponent (aExponent) modulo a given modulus (aModulus). It uses an iterative approach to efficiently calculate the result, avoiding potential overflows for large exponents.

In summary, this program generates and displays generalized Curzon numbers for various bases, providing a glimpse into these intriguing mathematical objects.

public final class CurzonNumbers {

	public static void main(String[] aArgs) {
		for ( int k = 2; k <= 10; k += 2 ) {
	        System.out.println("Generalised Curzon numbers with base " + k + ":");
	        int n = 1;
	        int count = 0;
	        while ( count < 50 ) {
	            if ( isGeneralisedCurzonNumber(k, n) ) {
	                System.out.print(String.format("%4d%s", n, ( ++count % 10 == 0 ? "\n" : " " )));
	            }
	            n += 1;
	        }
	        
	        while ( count < 1_000 ) {
	        	if ( isGeneralisedCurzonNumber(k, n) ) {
	        		count += 1;
	        	}
	        	n += 1;
	        }	 
	        System.out.println("1,000th Generalised Curzon number with base " + k + ": " + ( n - 1 ));
	        System.out.println();
	    }
	}
	
	private static boolean isGeneralisedCurzonNumber(int aK, int aN) {
	    final long r = aK * aN;
	    return modulusPower(aK, aN, r + 1) == r;
	}
	
	private static long modulusPower(long aBase, long aExponent, long aModulus) {
	    if ( aModulus == 1 ) {
	        return 0;
	    }	    
	    
	    aBase %= aModulus;
	    long result = 1;
	    while ( aExponent > 0 ) {
	        if ( ( aExponent  & 1 ) == 1 ) {
	            result = ( result * aBase ) % aModulus;
	        }
	        aBase = ( aBase * aBase ) % aModulus;
	        aExponent >>= 1;
	    }
	    return result;
	}

}