# All boils down to one

$\begin{array} \\ 3^2+4^2 &=& 5^2 \\ 5^2+12^2 &=& 13^2 \\ 6^2+8^2 &=& 10^2 \\7^2 + 24^2 &=& 25^2 \\ \end{array}$

A Pythagoras Triplet $$(A,B,C)$$ satisfy the condition that $$A,B,C$$ are positive integers such that $$A^2 + B^2 = C^2$$. From above, we can see that $$(3,4,5), (5,12,13), (6,8,10),(7,24,25)$$ are all Pythagoras Triplets.

Note that equations above are equivalent to

$\begin{array} \\ (\color{blue}2^2 - \color{green}1^2)^2 + (2 \cdot \color{blue}2 \cdot \color{green}1)^2 & = & (\color{blue}2^2 + \color{green}1^2)^2 \\ (\color{blue}3^2 - \color{green}2^2)^2 + (2 \cdot \color{blue}3 \cdot \color{green}2)^2 & = & (\color{blue}3^2 + \color{green}2^2)^2 \\ (\color{blue}3^2 - \color{green}1^2)^2 + (2 \cdot \color{blue}3 \cdot \color{green}1)^2 & = & (\color{blue}3^2 + \color{green}1^2)^2 \\ (\color{blue}4^2 - \color{green}3^2)^2 + (2 \cdot \color{blue}4 \cdot \color{green}3)^2 & = & (\color{blue}4^2 + \color{green}3^2)^2 \\ \end{array}$

True or false?:

"All Pythagoras Triplets $$(A,B,C)$$ can be written as $$(m^2 - n^2, 2mn, m^2 + n^2 )$$ for positive integers $$m,n$$ with $$m>n$$."

Note that the values of $$A$$ and $$B$$ are interexchangeable, so the triplet $$(3,4,5)$$ is equivalent to $$(4,3,5)$$.

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