Another Note on the Greatest Prime Factors of Fermat Numbers

Abstract

For every positive integer k > 1, let P(k) be the largest prime divisor of k. In this note, we show that if F _m = 2^2m + 1 is the m‘th Fermat number, then P(F _m ) ≥ 2^m+2(4m + 9) + 1 for all m ≥ 4. We also give a lower bound of a similar type for P(F _a,m ), where F _a,m = a^2m + 1 whenever a is even and m ≥ a¹⁸.