Lower bound for the greatest prime divisors of the generalized Fermat numbers (Q1769329)

For every positive integer \(k>1\), let \(P(k)\) be the largest prime divisor of \(k\). Denote by \(F_n=2^{2^n}+1\) the \(n\)th Fermat number and by \[ F_n(a)=a^{2^n}+1 \] the \(n\)the generalized Fermat number. Then \textit{M. Le} [Southeast Asian Bull. Math. 22, No.~1, 41--44 (1998; Zbl 0937.11001)] has shown that \(P(F_n(2))>2^{n-4}n\) if \(n\geq 2^{18}\) using \textit{P. Dong}'s result on linear forms in \(p\)-adic logarithms [C. R. Acad. Sci., Paris, Sér. I 315, No. 5, 503--506 (1992; Zbl 0774.11035)] and an effective bound of \textit{R. C. Vaughan} and \textit{H. L. Montgomery} for the number of primes in arithmetic progressions [Mathematika 20, 119--134 (1973; Zbl 0296.10023)]. By elementary arguments \textit{A. Grytczuk, M. Wójtowicz} and \textit{F. Luca} [Southeast Asian Bull. Math. 25 , No.~1, 111--115 (2001; Zbl 1009.11005)], improved this inequality to \(P(F_n(2))\geq (4n+9)2^{n+2}+1\) if \(n\geq 4\). In this paper, the authors prove the inequality \[ P(F_n(a))>\frac{2^{n+2}(n+1)\log 2} {\log a}+1 \] if \(n\geq 3\). The proof uses entirely elementary arguments from [A. Grytczuk et al., loc. cit.].