Mersenne Number

 

Write a program to check if a number is a Mersenne number or not.

In  mathematics, a Mersenne number is a number that can be written in the form M(n) = 2ⁿ − 1 for some integer n.
The first four Mersenne primes are 3, 7, 31, and 127

Test Data
Input a number: 127

Sample Solution:

Java Code:

import java.util.Scanner;
import java.math.BigInteger;
public class Example22 {

    public static void main(String args[])
    {
        Scanner in = new Scanner(System.in);
        System.out.print("Input a number: ");
        int n = in.nextInt();
        int n1 = n + 1;

        int power = 0;
        int ans = 0;
        for(int i=0;;i++)
        {
            power=(int)Math.pow(2,i);
            if(power>n1)
            {
                break;
            }
            else if(power==n1)
            {
               System.out.println(n+" is a Mersenne number.");
               ans=1;
            }
        }
  if(ans==0)
  {
   System.out.println(n+" is not a Mersenne number.");
  }
    }
}

Sample Output:

Input a number: 127                                                                                           
127 is a Mersenne number.

Ramanujan Numbers

 

Write a Java program to find any number between 1 and n that can be expressed as the sum of two cubes in two (or more) different ways.

Here are some examples of Ramanujan numbers :
1729 = 1^3 + 12^3 = 9^3 + 10^3
* 10000
1729 = 1^3 + 12^3 = 9^3 + 10^3
4104 = 2^3 + 16^3 = 9^3 + 15^3
* 100000
1729 = 1^3 + 12^3 = 9^3 + 10^3
4104 = 2^3 + 16^3 = 9^3 + 15^3
13832 = 2^3 + 24^3 = 18^3 + 20^3
39312 = 2^3 + 34^3 = 15^3 + 33^3
46683 = 3^3 + 36^3 = 27^3 + 30^3
32832 = 4^3 + 32^3 = 18^3 + 30^3
40033 = 9^3 + 34^3 = 16^3 + 33^3
20683 = 10^3 + 27^3 = 19^3 + 24^3
65728 = 12^3 + 40^3 = 31^3 + 33^3
64232 = 17^3 + 39^3 = 26^3 + 36^3

Sample Solution:

Java Code:

import java.util.Scanner;
public class Example21  {

   public static void main(String[] args) { 

         int n = 100000;
        // for each a, b, c, d, check whether a^3 + b^3 = c^3 + d^3
        for (int a = 1; a <= n; a++) {
            int a3 = a*a*a;
            if (a3 > n) break;

            for (int b = a; b <= n; b++) {
                int b3 = b*b*b;
                if (a3 + b3 > n) break;

                 for (int c = a + 1; c <= n; c++) {
                    int c3 = c*c*c;
                    if (c3 > a3 + b3) break;

                    for (int d = c; d <= n; d++) {
                        int d3 = d*d*d;
                        if (c3 + d3 > a3 + b3) break;

                        if (c3 + d3 == a3 + b3) {
                            System.out.print((a3+b3) + " = ");
                            System.out.print(a + "^3 + " + b + "^3 = "); 
                            System.out.print(c + "^3 + " + d + "^3"); 
                            System.out.println();
                        }
                    }
                }
            }
        }
    }
}

Sample Output:

1729 = 1^3 + 12^3 = 9^3 + 10^3                                                                                
4104 = 2^3 + 16^3 = 9^3 + 15^3                                                                                
13832 = 2^3 + 24^3 = 18^3 + 20^3                                                                              
39312 = 2^3 + 34^3 = 15^3 + 33^3                                                                              
46683 = 3^3 + 36^3 = 27^3 + 30^3                                                                              
32832 = 4^3 + 32^3 = 18^3 + 30^3                                                                              
40033 = 9^3 + 34^3 = 16^3 + 33^3                                                                              
20683 = 10^3 + 27^3 = 19^3 + 24^3                                                                             
65728 = 12^3 + 40^3 = 31^3 + 33^3                                                                             
64232 = 17^3 + 39^3 = 26^3 + 36^3

Cyclic Number

WAP to check a number is cyclic number or not

 class GFG {

    // Function to generate all cyclic

    // permutations of a number

    static boolean isCyclic(long N)

    {

        // Count digits and check if all

        // digits are same

        long num = N;

        int count = 0;

        int digit = (int)(num % 10);

        boolean allSame = true;

        while (num > 0) {

            count++;

            if (num % 10 != digit)

                allSame = false;

            num = num / 10;

        }

  

        // If all digits are same, then

        // not considered cyclic.

        if (allSame == true)

            return false;

  

        // If counts of digits is even and

        // two halves are same, then the

        // number is not considered cyclic.

        if (count % 2 == 0) {

            long halfPower = (long)Math.pow(10, count / 2);

            long firstHalf = N % halfPower;

            long secondHalf = N / halfPower;

            if (firstHalf == firstHalf && isCyclic(firstHalf))

                return false;

        }

  

        num = N;

        while (true) {

            // Following three lines generates a

            // circular permutation of a number.

            long rem = num % 10;

            long div = num / 10;

            num = (long)(Math.pow(10, count - 1))

                      * rem

                  + div;

  

            // If all the permutations are checked

            // and we obtain original number exit

            // from loop.

            if (num == N)

                break;

  

            if (num % N != 0)

                return false;

        }

  

        return true;

    }

  

    // Driver code

    public static void main(String[] args)

    {

        long N = 142857;

        if (isCyclic(N))

            System.out.print("Yes");

        else

            System.out.print("No");

    }

}