$x^2+1$ 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?

It is not true that $x^2+1$ is almost always square-free. For if $x\equiv 7\mod{25}$, then $x^2+1$ is divisible by $25$.
