Another note on the greatest prime factors of Fermat numbers (Q5943067)

For any positive integer \(a\) with \(a>1\), let \(P(a)\) denote the greatest prime factor of \(a\). For any positive integers \(b\) and \(m\), let \(F_{b,m}= b^{2^m}+1\) be the generalized Fermat number. When \(b=2\), \(F_{2,m}= F_m\) is the usual Fermat number. Using the Gel'fond-Baker method, the reviewer [Southeast Asian Bull. Math. 22, 41-44 (1998; Zbl 0937.11001)] proved that if \(m\geq 2^{18}\), then \(P(F_m)> 2^{m-4}m\). In this paper, using an elementary argument, the authors prove that if \(m\geq 4\), then \(P(F_m)\geq 2^{m+2} (4m+9)+1\). On the other hand, by the reviewer's argument, they prove that if \(m\geq b^{18}\), then \(P(F_{b,m})> 2^{m-4}m\).