[Khoth]

Consider Goldbach's Conjecture ( https://en.wikipedia.org/wiki/Goldbach%27s_conjecture ), which states that every even number greater than 2 is the sum of two prime numbers.

You could write quite a short program to search for a counterexample - for each even number, check to see if it's the sum of two prime numbers. If it's not, then print it and halt, if it is then go on to the next number.

Looking at this program, you could easily understand how it works, but to know if it halts you need to solve one of the most famous open problems in mathematics.

[Tainnor]

That's not directly related. The halting problem states that we cannot write a program that can decide whether arbitrary programs halt - even if we had solved every mathematical question that is still open.