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

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.