<s>[INST] Bounds in Pythagorean Triples Let's take a look at this simple task (Pythagorean Triples):
Calculate $A$ and $B$ such that $A^2 + B^2 = C^2$.
$C$ is given.
Is there any way to find an upper bound for $A$, $B$, $A^2$, and $B^2$?
The upper bound will be a function of $C$.
$0 < A < f_1(C)$
$0 < B < f_1(C)$
$0 < A^2 < f_2(C)$
$0 < B^2 < f_2(C)$
$f_1(C) = ?$
$f_2(C) = ?$
Any ideas?
 [/INST] There are an infinite number of pythagorean triples for which $B + 1 = C$.  Let $A = 2n+1$.  Let $B = 2n^2 + 2n$ and $C = B + 1 = 2n^2 + 2n + 1$.
$$\begin{align}
\\ A^2 + B^2 &= (2n+1)^2 + (2n^2 + 2n)^2
\\ & = (4n^2 + 4n + 1) + (4n^4 + 8n^3 + 4n^2)
\\ & = 4n^4 + 8n^3 + 8n^2 + 4n + 1
\\ & = (2n^2 + 2n + 1)^2
\\ & = C^2
\end{align}
$$
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