An improvement on the largest prime factor of $n^2+1$

The study of the largest prime factor in polynomial sequences can be traced back at least to the late 19th century in the work of Störmer. Mahler (1933) and Chowla (1934) proved that the largest prime factor of $n^2+1$ grows at least as fast as ${\log }_{2}n$. In 2023 we improved this to $({\log }_{2}n{)}^2\mathrm{/}{\log }_{3}n$. In this note we show the lower bound $({\log }_{2}n{)}^2\mathrm{/}{\log }_{4}n$, and that when this bound is nearly sharp it also holds for many prime factors of $n^2+1$.