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.