Factors of Fermat numbers and large primes of the form 𝑘⋅2ⁿ+1

Author:

Keller Wilfrid

Abstract

A new factor is given for each of the Fermat numbers F 52 , F 931 , F 6835 {F_{52}},{F_{931}},{F_{6835}} , and F 9448 {F_{9448}} . In addition, a factor of F 75 {F_{75}} discovered by Gary Gostin is presented. The current status for all F m {F_m} is shown in a table. Primes of the form k 2 n + 1 , k k \cdot {2^n} + 1,k odd, are listed for 31 k 149 31 \leqslant k \leqslant 149 , 1500 > n 4000 1500 > n \leqslant 4000 , and for 151 k 199 151 \leqslant k \leqslant 199 , 1000 > n 4000 1000 > n \leqslant 4000 . Some primes for even larger values of n are included, the largest one being 5 2 13165 + 1 5 \cdot {2^{13165}} + 1 . Also, a survey of several related questions is given. In particular, values of k such that k 2 n + 1 k\cdot {2^n} + 1 is composite for every n are considered, as well as odd values of h such that 3 h 2 n ± 1 3h\cdot {2^n} \pm 1 never yields a twin prime pair.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,Computational Mathematics,Algebra and Number Theory

Reference24 articles.

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