The halting problem is a poor example as there is already a simple proof showing that such a solution does not exist. A much better example is P=NP. This is a

The halting problem is a poor example as there is already a simple proof showing that such a solution does not exist. A much better example is P=NP. This is a very rich area of research with many closely related problems that has been worked on for decades and will continue to be worked on for the foreseeable future.