(Ramanujan's Taxi) Srinivasa Ramanujan indian mathematician who became famous for his intuition for numbers. When the English mathemematician G. H. Hardy came ot visit him in the hospital one day, Hardy remarked that the number if his taxi was 1729, a rather dull number. To which Ramanujan replied, "No, Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways." Verify this claim by writing a program Ramanujan. java that takes a command-line argument N and prints out all integers less than or equal to N that can be expressed as the sum of two cubes in two different ways. In other words, find distinct positive integers a, b, c, and d such that а3 + b3 = c3 - d3. Hint: Use four nested for loops, with these bounds on the loop variables: 0 < a < 3√N, a < b < 3√N - a3, a < c < 3√N and c < d < 3√N - C3; do not explicitly compute cube roots, and instead use x . x *x < y in place of х < Math. cbrt (y). $ javac Ramanujan. java
$ java Ramanujan 40000
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
32832 = 4^3 + 32^3 = 18^3 + 30^3
20683 = 10^3 + 27^3 = 19^3 + 24^3

Here the code is given as follows,

Explanation:

Code:

public class Ramanujan {  

   public static void main(String[] args) {  

       // check cmd argument

       int n = Integer.parseInt(args[0]);  

       // checking a^3 + b^3 = c^3 + d^3  

       // outer loop

       for (int a = 1; (a*a*a) <= n; a++) {  

           int a3 = a*a*a;  

           // no solution is possible, since upcoming a values are more

           if (a3 > n)  

               break;  

           // avoid duplicate

           for (int b = a; (b * b * b) <= (n - a3); b++) {  

               int b3 = b*b*b;  

               if (a3 + b3 > n)

                   break;  

               // avoid duplicates

               for (int c = a + 1; (c*c*c) <= n; c++) {  

                   int c3 = c*c*c;  

                   if (c3 > a3 + b3)

                       break;  

                   // avoid duplicates

                   for (int d = c; (d * d * d) <= (n - c3); 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();

                       }

                   }

               }

           }

       }

   }

}

The main part about version control is that you have a repository, which hosts files. any changes done to those files are recorded and, as such, can be reverted back. it is kind of like google drive/google docs (except not as "real-time autosave"). usually these files involve code. version control can usually record changes to files and the lines of code themselves.version control is used by developers, mainly with coding. this may include websites, software, video games, etc. it is very useful in terms of practicality and collaboration.  reverting back a previous version of a file is very useful in the coding/programming world. suppose a programmer adds in something new to a file, a code snippet or two. then he changes his mind because he does not like the feature he coded in (it does not work, inefficient, with version control, he can easily revert back the changes he made.  version control also makes it easier to collaborate. oftentimes, you can use an online, remote  repository such as github or bitbucket. a team of people  can each  connect their computers  to this repository and be synchronised with the source code.recording your changes is also for teamwork  as it tells you who did what. changes can be monitored as well.  online repositories (like  github) may also contain  features to talk about issues with the code, and you can  reference them in the changes.version control also allows you to have different versions of the same thing. there are open source projects where anyone able can contribute to the project; such collaborators are able to create a new version (a new "branch") from the master branch. in this new branch derived from the main master branch, one adds their code and such.  in time, the original developer will look at this branch; they can add all or some of those changes to the main branch if they approve.