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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Factors of Fermat numbers and large primes of the form $ k\cdot 2\sp{n}+1$

Author: Wilfrid Keller
Journal: Math. Comp. 41 (1983), 661-673
MSC: Primary 11Y05; Secondary 11A41, 11Y11
MathSciNet review: 717710
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Abstract: A new factor is given for each of the Fermat numbers $ {F_{52}},{F_{931}},{F_{6835}}$, and $ {F_{9448}}$. In addition, a factor of $ {F_{75}}$ discovered by Gary Gostin is presented. The current status for all $ {F_m}$ is shown in a table. Primes of the form $ k \cdot {2^n} + 1,k$ odd, are listed for $ 31 \leqslant k \leqslant 149$, $ 1500 < n \leqslant 4000$, and for $ 151 \leqslant k \leqslant 199$, $ 1000 < n \leqslant 4000$. Some primes for even larger values of n are included, the largest one being $ 5 \cdot {2^{13165}} + 1$. Also, a survey of several related questions is given. In particular, values of k such that $ k\cdot{2^n} + 1$ is composite for every n are considered, as well as odd values of h such that $ 3h\cdot{2^n} \pm 1$ never yields a twin prime pair.

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Keywords: Fermat numbers, numbers of Ferentinou-Nicolacopoulou, factoring, trial division, large primes, covering set of divisors, twin primes
Article copyright: © Copyright 1983 American Mathematical Society

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