Let $H_n$ be the hypotenuse of the $n$-th primitive Pythagorean Triplet when arranged in ascending order of the length of the hypotenuse. What is known about the asymptotic expansion of or bounds on $H_n$?
My experimental data suggests that
$$ 2\pi n - 4n^{\frac{1}{3}} - 4 < H_n < 2\pi n + 4n^{\frac{1}{3}} + 4 $$
There are no violations of this inequality for $n \le 1.4 \times 10^8$.
A graph of hypotenuse values shows that they are very very close to $\space 2\pi n.\space $ There are $\space 2^{−1}\space $ primitive triples for every $\space _\space $ where $\space \space $ is the number of distinct prime factors of $\space _. \space$ If you allow for these multiple triples for some $\space _\space $ values: $\quad (6-1)\le H_n \le (7-1)\quad$