S(n) = n^n + 1

Can the number S(n) = n^n+1 be prime? It is easy to show that a necessary condition is that n is of the form n=2^(2^k). The number S(n) therefore becomes a Fermat number F(m)=2^(2^m)+1 where m=k+2^k (Sloane's A006127).

Here is the status of such numbers. It seems extremely unlikely that any more of the numbers S(n) would be prime (A121270).

k m n status of S(n)
0 1 2 prime, 2^2 + 1 = 5
1 3 4 prime, 4^4 + 1 = 257
2 6 16 composite, 16^16 + 1 has factor 1071*2^8 + 1
3 11 256 composite, 256^256 + 1 has factor 39*2^13 + 1
4 20 65536 composite, but no factor known
5 37 composite, S(n) has factor 1275438465*2^39 + 1
6 70 status unknown
7 135 status unknown
8 264 status unknown
9 521 status unknown
10 1034 status unknown
11 2059 composite, S(n) has factor 591909*2^2063 + 1
12 4108 status unknown
13 8205 status unknown
14 16398 status unknown
15 32783 status unknown
16 65552 status unknown
17 131089 status unknown
18 262162 status unknown
19 524307 status unknown
20 1048596 status unknown
21 2097173 status unknown
22 4194326 status unknown
23 8388631 status unknown
24 16777240 status unknown
25 33554457 status unknown
26 67108890 status unknown

k	m	n	status of S(n)
0	1	2	prime, 2^2 + 1 = 5
1	3	4	prime, 4^4 + 1 = 257
2	6	16	composite, 16^16 + 1 has factor 1071*2^8 + 1
3	11	256	composite, 256^256 + 1 has factor 39*2^13 + 1
4	20	65536	composite, but no factor known
5	37		composite, S(n) has factor 1275438465*2^39 + 1
6	70		status unknown
7	135		status unknown
8	264		status unknown
9	521		status unknown
10	1034		status unknown
11	2059		composite, S(n) has factor 591909*2^2063 + 1
12	4108		status unknown
13	8205		status unknown
14	16398		status unknown
15	32783		status unknown
16	65552		status unknown
17	131089		status unknown
18	262162		status unknown
19	524307		status unknown
20	1048596		status unknown
21	2097173		status unknown
22	4194326		status unknown
23	8388631		status unknown
24	16777240		status unknown
25	33554457		status unknown
26	67108890		status unknown

Links: Weisstein, Keller.

/JeppeSN