Prime Sierpiński Numbers of the First Kind

Open Question
The only known Sierpiński primes of the first kind are:

It is an open question as to whether there are any more.

Progress
Let $S_n$ be the $n$th Sierpiński number of the first kind:
 * $S_n = n^n + 1$

From Form of Sierpiński Prime of the First Kind, it is known that if $S_n$ is prime, then:
 * $n = 2^{2^k}$

for some integer $k$.

Thus

Thus a prime $S_n$ is a Fermat number $F_m$ where $m = k + 2^k$.

The sequence of $m$ begins:
 * $1, 3, 6, 11, 20, 37, 70, 135, 264, 521, 1034, 2059, 4108, 8205, \ldots$

Thus it remains to investigate the following Sierpiński numbers of the first kind for primality:


 * {| border="1"

! align="right" style = "padding: 2px 10px" | $k$ ! align="right" style = "padding: 2px 10px" | $m = k + 2^k$ ! align="right" style = "padding: 2px 10px" | $n = 2^{2^k}$ ! style = "padding: 2px 10px" | Status of $S_n = n^n + 1 = 2^{2^m} + 1 = F_m$
 * align="right" style = "padding: 2px 10px" | $0$
 * align="right" style = "padding: 2px 10px" | $1$
 * align="right" style = "padding: 2px 10px" | $2$
 * style = "padding: 2px 10px" | Prime ($S_2 = F_1 = 5$)
 * align="right" style = "padding: 2px 10px" | $1$
 * align="right" style = "padding: 2px 10px" | $3$
 * align="right" style = "padding: 2px 10px" | $4$
 * style = "padding: 2px 10px" | Prime ($S_4 = F_3 = 257$)
 * align="right" style = "padding: 2px 10px" | $2$
 * align="right" style = "padding: 2px 10px" | $6$
 * align="right" style = "padding: 2px 10px" | $16$
 * style = "padding: 2px 10px" | Composite with factor $1071 \times 2^8 + 1$
 * align="right" style = "padding: 2px 10px" | $3$
 * align="right" style = "padding: 2px 10px" | $11$
 * align="right" style = "padding: 2px 10px" | $2^{2048}$
 * style = "padding: 2px 10px" | Composite with factor $39 \times 2^{13} + 1$
 * align="right" style = "padding: 2px 10px" | $4$
 * align="right" style = "padding: 2px 10px" | $20$
 * align="right" style = "padding: 2px 10px" | $2^{1 \, 048 \, 576}$
 * style = "padding: 2px 10px" | Composite with no factor known
 * align="right" style = "padding: 2px 10px" | $5$
 * align="right" style = "padding: 2px 10px" | $37$
 * align="right" style = "padding: 2px 10px" | too large
 * style = "padding: 2px 10px" | Composite with factor $1 \, 275 \, 438 \, 465 \times 2^{39} + 1$
 * align="right" style = "padding: 2px 10px" | $6$
 * align="right" style = "padding: 2px 10px" | $70$
 * align="right" style = "padding: 2px 10px" |
 * style = "padding: 2px 10px" | unknown
 * align="right" style = "padding: 2px 10px" | $7$
 * align="right" style = "padding: 2px 10px" | $135$
 * align="right" style = "padding: 2px 10px" |
 * style = "padding: 2px 10px" | unknown
 * align="right" style = "padding: 2px 10px" | $8$
 * align="right" style = "padding: 2px 10px" | $264$
 * align="right" style = "padding: 2px 10px" |
 * style = "padding: 2px 10px" | unknown
 * align="right" style = "padding: 2px 10px" | $9$
 * align="right" style = "padding: 2px 10px" | $521$
 * align="right" style = "padding: 2px 10px" |
 * style = "padding: 2px 10px" | unknown
 * align="right" style = "padding: 2px 10px" | $10$
 * align="right" style = "padding: 2px 10px" | $1034$
 * align="right" style = "padding: 2px 10px" |
 * style = "padding: 2px 10px" | unknown
 * align="right" style = "padding: 2px 10px" | $11$
 * align="right" style = "padding: 2px 10px" | $2059$
 * align="right" style = "padding: 2px 10px" |
 * style = "padding: 2px 10px" | Composite with factor $591 \, 909 \times 2^{2063} + 1$
 * align="right" style = "padding: 2px 10px" | $12$
 * align="right" style = "padding: 2px 10px" | $4108$
 * align="right" style = "padding: 2px 10px" |
 * style = "padding: 2px 10px" | unknown
 * align="right" style = "padding: 2px 10px" | $13$
 * align="right" style = "padding: 2px 10px" | $8205$
 * align="right" style = "padding: 2px 10px" |
 * style = "padding: 2px 10px" | unknown
 * align="right" style = "padding: 2px 10px" | $14$
 * align="right" style = "padding: 2px 10px" | $16 \, 398$
 * align="right" style = "padding: 2px 10px" |
 * style = "padding: 2px 10px" | unknown
 * align="right" style = "padding: 2px 10px" | $15$
 * align="right" style = "padding: 2px 10px" | $32 \, 783$
 * align="right" style = "padding: 2px 10px" |
 * style = "padding: 2px 10px" | unknown
 * align="right" style = "padding: 2px 10px" | $16$
 * align="right" style = "padding: 2px 10px" | $65 \, 552$
 * align="right" style = "padding: 2px 10px" |
 * style = "padding: 2px 10px" | unknown
 * align="right" style = "padding: 2px 10px" | $17$
 * align="right" style = "padding: 2px 10px" | $131 \, 089$
 * align="right" style = "padding: 2px 10px" |
 * style = "padding: 2px 10px" | unknown
 * }
 * align="right" style = "padding: 2px 10px" |
 * style = "padding: 2px 10px" | unknown
 * align="right" style = "padding: 2px 10px" | $15$
 * align="right" style = "padding: 2px 10px" | $32 \, 783$
 * align="right" style = "padding: 2px 10px" |
 * style = "padding: 2px 10px" | unknown
 * align="right" style = "padding: 2px 10px" | $16$
 * align="right" style = "padding: 2px 10px" | $65 \, 552$
 * align="right" style = "padding: 2px 10px" |
 * style = "padding: 2px 10px" | unknown
 * align="right" style = "padding: 2px 10px" | $17$
 * align="right" style = "padding: 2px 10px" | $131 \, 089$
 * align="right" style = "padding: 2px 10px" |
 * style = "padding: 2px 10px" | unknown
 * }
 * align="right" style = "padding: 2px 10px" |
 * style = "padding: 2px 10px" | unknown
 * }