As The Masked Avenger has hinted at, one can find all $n$ for which $n^2+1$ has all prime factors below a given bound by looking among the first several solutions to finitely many Pell equations. Størmer described this method already in 1897. This predates Siegel by thirty years. See also <p>Lehmer, D. H. (1964). "On a Problem of Størmer". Illinois Journal of Mathematics 8: 57–79. MR 0158849.

(Note that Størmer's method is less pliable than the solutions based on Siegel's theorem; it applies only to $n^2+1$ and a few other polynomials.)

For a modern variant of Størmer, see e.g.,
http://web.math.pmf.unizg.hr/glasnik/45.2/45(2)-04.pdf