This is a simple(it look simple!) math puzzle which suddenly came to my mind:
How many different answers does 2m + 2n = k have? m,n and k are positive integers and k is a constant.
For example for k=10 the equation have 2 answers(23 + 21 and vice versa) and for k=3 or any odd number we have zero answers.finally for any positive power of 2(except 1) we have just one answer.
If you found any thing please email me and prove any relation which you find.