Distribution of Primitive Pythagorean Triples (PPT) and of solutions of $A^4+B^4+C^4=D^4$ If we define a $PPTCountingFunction(n)$ as a function that returns the number of PPF with $c < n$ and $a>b$, then up to first $n=100,000$ it is near linear and   
$\dfrac{n}{PPTCountingFunction(n)}=2\pi$
I have several questions (third question is the most interesting to me):
(1) Is this also an asymptotic behavior of this function, or does it have some other slowly changing factors that are not showing up when n is small?  
(2) Is there a clear reasoning for frequencies of PPT?   
(3) Can we apply similar reasoning to estimate the frequency of primitive counterexamples to Euler's hypothesis for $n=4$ (solution s of $A^{4}+B^{4}+C^{4}=D^{4}$)?

Regarding (3). First solution appears at $95800^{4} + 414560^{4} + 217519^{4} = 422481^{4}$. This is the only solution with $D<2000000$. Another known solution (not necessarily second)  is  $2682440^{4}  +  15365639^{4}  +  18796760^{4}  =  20615673^{4}$. I am curious if there is a point to look for a solution between these two.
 A: The list of hypotenuses is A020882 in OEIS. The following analysis follows the third comment on that sequence, and provides a reasonability argument, though not a proof. 
Counting primitive triples with hypotenuse at most $n$ is the same as counting pairs $(a,b)$ with $\gcd(a,b)=1$, $a$ and $b$ not both odd, and $a>b>0$ inside the circle of radius $\sqrt{n}$. The total number of pairs $(a,b)$ inside such a circle is $\approx \pi n$ (see here, for example). Only $\frac{1}{8}$ of these have $a>b>0$; of these, only $\frac{6}{\pi^2}$ of them are relatively prime. Finally, asking that $a$ and $b$ be not both odd reduces by another factor of $\frac{2}{3}$ (note that $a$, $b$ both even was excluded by the $\gcd$). So altogether, the number of qualifying points is
$$n\cdot\pi\cdot\frac{1}{8}\cdot\frac{6}{\pi^2}\cdot\frac{2}{3} = \frac{n}{2\pi}.$$
Thus $\text{ppt}(n)\approx \frac{n}{2\pi}$, and your result follows.
A: In response to (1), this behaviour of the function is indeed asymptotic.  This was proved in D N Lehmer (1900) Asymptotic Evaluation of Certain Totient Sums American Journal of Mathematics Vol 22, freely available via JSTOR here.  This result (although the term 'Pythagorean triple' is not used) is on pp 327-8.  
