Another Note on the Greatest Prime Factors of Fermat Numbers
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 = 22m + 1 is the m‘th Fermat number, then P(F m ) ≥ 2m+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 = a2m + 1 whenever a is even and m ≥ a18.
Keywords.Fermat number greatest prime factor linear forms in p-adic logarithms
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