It is now proved that, for integer $n\geq 2$, the equation $x^n+y^n=z^n$ has integer solution only when $n=2$.
When $n=2$, this equation has an infinity of solutions.
My question is whether there is some established knowledge on the density of solutions and its variation in the space of integer pairs $(x,y)$ or $(x,z)$.
The following was added after reading the first answer by 01000100 (thanks).
The Lehmer asymptotic evaluation does give some indication of what I am after. But it is limited. If, in the $(x,y)$ plane, all pythagorean triples were concentrated around the $x=y$ line (the first diagonal), none being near the horizontal and vertical axes, it could follow the Lehmer density result, though be far from asymptotically uniform in relative density.
So I would rather consider something more or less like the following (which may not be quite correct a way to express what I have in mind):
Let $P(m,n)$ be the number of primitive Pythagorean triples $(x,y,z)$ with $m/2\leq x\leq m$ and $n/2\leq y\leq n$, with $gcd(x,y,z)=1$. What is the limit $$\lim_{p \rightarrow \infty} \frac{P(ap,bp)}{abp^2/4}$$ where $a,b,p\in \mathbb N$? Note that $abp^2/4$ is simply the surface of the rectangle where primitive Pythagorean triples are considered for $P(ap,bp)$.
This is intended just to control the plane direction in which the density is measured. But one could consider more complex relations between the two parameters of $P$.
Or another question (which I think is not equivalent, but I am not sure):
Is the set $\{x/y\mid x^2+y^2=z^2 \wedge (x,y,z)\in \mathbb N^3\}$ dense in the reals?
And I have similar questions regarding $(x,z)$ rather than $(x,y)$.
First, it's a little silly to call this "Fermat's equation".
Now, points with both coordinates rational are dense on the unit circle; any line through $(1,0)$ with rational slope hits the circle in such a point. This should enable you to answer your questions about density.