Density of Pythagorean triples We define a Pythagorean triple as a triple $<a,b,c>$ such that $a,b,c\in \mathbb N$ and $a^2+b^2=c^2$.
In order to avoid duplicates, we say that a triple $<a,b,c>$ is legit iff $b>a$.

Let $\mathcal P$ be the set of all legit Pythagorean triples.
We define $$L_{PT}^N=\{<a,b,c> | <a,b,c> \in  \mathcal P\wedge b\leq N\}$$
(If it's more convinient we can define it for $b^2\leq N$, $c\leq N$ or $c^2\leq N$).

What is the density of $|L_{PT}^N|$ as a function of $N$? e.g. is $|L_{PT}^N|=\Theta(N^2)?\Theta(N)?$


We say that a triple $<a,b,c>$ is minimal if $gcd(a,b,c)=1$.
Let $\mathcal P_M$ be the set of all legit, minimal triples.
Let $$L_{MPT}^N=\{<a,b,c> | <a,b,c> \in  \mathcal P_M\wedge b\leq N\}$$

What is the density of $|L_{MPT}^N|$ as a function of $N$? e.g. is $|L_{MPT}^N|=\Theta(N)?$


It was proved by Lehmer that the number of primitive triples with hypotenuse less than $x$ is asymptotic to
$$
\frac{x}{2\pi} \approx 0.15915494309x
$$
