$x^2+1$ is almost always square free It seems like $x^2+1$ is almost always square free. Any research or heuristics why?
I tried breaking the problem into solving $$x^2-ky^2=1$$
For various $k$, and I conjecture that for every $k$ there are at most finitely many solutions to the Diophantine equation. I'm pretty sure this is correct but dont know how to prove it. Any ideas on  my two problems?

The equation
$$x^2-ky^2=1$$
for non-square $k$ is known as Pell's equation. Lagrange proved that every Pell's equation has infinitely many solutions. Moreover, if $(x_0,y_0)$ is the smallest non-trivial positive solution, all integer solutions are implicitly given by
$$x+y\sqrt k=\pm\,(x_0+y_0\sqrt k)^n\;, \qquad n\in\mathbb Z.$$
As you can see, the solutions generated by this formula quickly become very large, making it natural to conjecture that there are only finitely many solutions. For $k=991$, the smallest solution is
$$y=12,055,735,790,331,359,447,442,538,767,$$
which illustrates that the smallest solution can still be extremely big.
(found here, actually Joseph Rotman's A First Course in Albegra: with applications)
Note that for square $k=m^2$ the equation becomes $(x+my)(x-my)=1$, which has at most $2$ solutions.
Regarding the square-freeness of $x^2+1$, I found this paper by D.R. Heath-Brown on arXiv, giving asymptotics for the number of $x$ for which $x^2+1$ is square-free. It turns out that such numbers have non-zero asymptotic density equal to
$$\frac12\prod_{p\equiv1\pmod4}\left(1-\frac2{p^2}\right).$$
