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Here is the claim of the proof:

If m is a positive integar of the form m=2s, where s is an odd integar, then there do not exist positive integars x and y such that x^2-y^2=m.

I am supposed to write a proof to this problem using proof by contradiction.

It is known that x²-y² factors into (x-y)(x+y).

Since x and y are integers, there are three approaches:

Both are odd, 1 is odd and 1 is even, or both are even.

If both are even or both are odd, then x-y and x+y are both even.

This means that multiplying them together results in a number divisible by 4, not just 2.

If one is odd and the other even, then x-y and x+y are both odd.

Since neither is divisible by 2, the product is not divisible by 2 either.

From the two previous arguments, we can conclude that x²-y² is divisible by 2 at least twice or none at all.

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