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.