All boils down to one

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

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

$\begin{array}{c}\\ (\color{#3D99F6}2^2 - \color{#20A900}1^2)^2 + (2 \cdot \color{#3D99F6}2 \cdot \color{#20A900}1)^2 & = & (\color{#3D99F6}2^2 + \color{#20A900}1^2)^2 \\ (\color{#3D99F6}3^2 - \color{#20A900}2^2)^2 + (2 \cdot \color{#3D99F6}3 \cdot \color{#20A900}2)^2 & = & (\color{#3D99F6}3^2 + \color{#20A900}2^2)^2 \\ (\color{#3D99F6}3^2 - \color{#20A900}1^2)^2 + (2 \cdot \color{#3D99F6}3 \cdot \color{#20A900}1)^2 & = & (\color{#3D99F6}3^2 + \color{#20A900}1^2)^2 \\ (\color{#3D99F6}4^2 - \color{#20A900}3^2)^2 + (2 \cdot \color{#3D99F6}4 \cdot \color{#20A900}3)^2 & = & (\color{#3D99F6}4^2 + \color{#20A900}3^2)^2 \\ \end{array}$

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 $(4,3,5)$ .