Another Note on the Greatest Prime Factors of Fermat Numbers |
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Authors: | A. Grytczuk M. Wójtowicz F. Luca |
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Affiliation: | (1) Institute of Mathematics, T. Kotarbiski Pedagogical University, 65-069 Zielona Góra, Pl. Slowiaski 9, Poland;(2) Institute of Mathematics, Czech Academy of Sciences, itná 25, 115 67 Praha 1, Czech Republic |
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Abstract: | For every positive integer k > 1, let P(k) be the largest prime divisor of k. In this note, we show that if Fm = 22m + 1 is the mth Fermat number, then P(Fm) 2m+2(4m + 9) + 1 for all m 4. We also give a lower bound of a similar type for P(Fa,m), where Fa,m = a2m + 1 whenever a is even and m a18.AMS Subject Classification (1991) 11A51 11J86 |
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Keywords: | Fermat number greatest prime factor linear forms in p-adic logarithms |
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