Trivial but cool

Claim: Every odd integer square is an element of some Pythagorean triple.
We first show that every positive odd number 2n+1 is equivalent to the expression (n+1)² - n².
We have (n+1)² - n² = (n+1+n)(n+1-n) = 2n+1.
An arbitrary odd square m² = 2n+1 = (n+1)² - n².
That is, m² = (n+1)² - n² or
m² + n² = (n+1)².

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Note that we may let n+1 = k, for k² - (k-1)²
= (k+k-1)(k- k- 1) = -(2k -1) = 1 - 2k.

So, expressed this way, we obtain all the negative odd numbers.

We see this by

2n + 1 = 1 - 2k

or 2n = -2k

or n = -k.