Two problems on the greatest prime factor of n2+1

Let P+(m) denote the greatest prime factor of the positive integer m. In [Arch. Math. (Basel) 90 (2008), 239-245] we improved work of Dartyge [Acta Math. Hungar. 72 (1996), 1-34] to show that |{n <= x : P+(n(2) + 1) < x(alpha)}| >> x for alpha > 4/5. In this note we show how the recent work of de la Bret & egrave;che and Drappeau [J. Eur. Math. Soc. 22 (2020), 1577-1624] (which uses the improved bound for the smallest eigenvalue in the Ramanujan-Selb erg conjecture given by Kim [J. Amer. Math. Soc. 16 (2003), 139-183]) along with a change of argument can be used to reduce the exponent to 0.567. We also show how recent work of Merikoski [J. Eur. Math. Soc. 25 (2023), 1253-1284] on large values of P+(n(2) + 1) can improve work by Everest and the author [London Math. Soc. Lecture Note Ser. 352, Cambridge Univ. Press, 2008, 142-154] on primitive divisors of the sequence n(2) + 1.