Two problems on the greatest prime factor of
Tom 213 / 2024
Streszczenie
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 \le x: P^+(n^2+1) \lt x^{\alpha } \}| \gg x for \alpha \gt 4/5. In this note we show how the recent work of de la Bretèche and Drappeau [J. Eur. Math. Soc. 22 (2020), 1577–1624] (which uses the improved bound for the smallest eigenvalue in the Ramanujan–Selberg 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.