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I've been working on a proof, and have access to an answer in the back of the textbook, but am not completely satisfied with it, so I was hoping someone could help me work through the logic. The problem is as follows:
Statement: For any odd integer, a, there are natural numbers b and c such that a^2 +b^2 = c^2.
The book then gives several examples where this holds and asks that I prove this statement with a construction based on the examples, which are:
3^2 + 4^2 = 5^2
5^2 + 12^2 = 13^2
7^2 + 24^2 = 25^2
9^2 + 40^2 = 41^2
The answer in the back is:
(2r + 1)^2 + X^2 = (X+1)^2
where X = 2r^2 + 2r
I'm having trouble figuring out where this comes from. I know that, since the 'a' term above is odd, it can be expressed as (2r + 1) or (2r - 1), so the first term makes sense. I also see the relation between the 'b' and 'c' terms for some given value of 'b.' What I cannot see is the connection between 'a' and 'b', i.e., where the above expression for X comes from. I know that I can't simply pick any arbitrary odd number or 'b' term, but I can't understand how I would arrive at this.
Any help or even hints would be greatly appreciated. Thanks in advance.
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