First, we wish to show that . Let . Then or .
Case 1:
so that
so that
and so that
Case 2:
and
so that
so that
and so that
Since in both cases, , we know that
Now we wish to show that . Let . Then and .
Case 1a:
, so
Case 1b:
We can't actually conclude anything we want with just this, so we have to also to consider the case .
Case 2a: : [see Case 1a]
Case 2b:
We now have and so that
Of course, since , it follows that .
Since both cases 2a and 2b yield , we know that it follows from 1b.
Since in both cases 1a and 1b, , we know that .
Since both and , it follows (finally) that .
--will continue later, feel free to refine it if you feel it can be--
- Because the question asks about the square of a number, you can substitute the definition of an odd number 2n + 1 into the number to be squared. So, say x is that number, then
- Multiply both factors together
- Factor out a two for the first two terms
- The factor will always be a natural number. As such, it fits the definition of an odd number, 2n + 1
- Problem solved!