A student asked me the other day the following rather homework-looking question: Given a natural number , how many solutions does the equation
have for and natural numbers?
The question has a very easy answer: Simply notice that and that any like this determines a unique such that is a solution. So, there are solutions if is even (as can be any of ), and there are solutions if is odd.
I didn’t tell the student what the answer is, but I asked what he had tried so far. Among what he showed me there was a piece of paper in which somebody else had scribbled
which caught my interest, and is the reason for this posting.