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. St"{o}rmer described this method already in 1897. This predates Siegel by thirty years. See also

Lehmer, D. H. (1964). "On a Problem of Størmer". Illinois Journal of Mathematics 8: 57–79. MR 0158849.

(Note that St"{o}rmer'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 St"{o}rmer, see e.g., http://web.math.pmf.unizg.hr/glasnik/45.2/45(2)-04.pdf

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

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