Are there infinite many integer $n\ge 0$ such that $10^{2^n}+1$ prime numbers?

It is clear to see that 11 and 101 are primes which sum of digit is 2. I wonder are there more or infinte many of such prime.

At first, I was think of the number $10^n+1$. Soon, I knew that $n\neq km$ for odd $k>1$, otherwise $10^m+1$ is a factor.

So, here is my question:

Are there infinite many integer $n\ge 0$ such that $10^{2^n}+1$ prime numbers?

After a few minutes: I found that if $n=2$, $10^{2^n}+1=10001=73\times137$, not a prime; if $n=3$, $10^{2^n}+1=17\times5882353$, not a prime; $n=4$, $10^{2^n}+1=353\times449\times641\times1409\times69857$, not a prime.

Now I wonder if 11 and 101 are the only two primes with this property.

Many people wonder the same thing you do. Wilfrid Keller keeps track of what they find out. So far: prime for $n=0$ and $n=1$ only; known to be composite for all other $n$, $2\le n\le23$, and many other values of $n$. The first value for which primality status is unknown is $n=24$.

Since no one else has mentioned it:

Standard heuristics in number theory suggest that there are only finitely many primes of the form $(2k)^{2^n}+1$ for any integer $k>0.$ The probability that a random number around $(2k)^{2^n}+1$ is prime is roughly $1/(2^n\log(2k))$; if you take into account the congruence conditions for such numbers and treat the chance that such a number is prime as a random variable, then the expectation is $C_k/2^n$ and the sum over these values converges.

If you sum this 'probability' over $n\ge24$ the expected number of primes of this form is less than 0.000001.

If you're interested in quickly determining whether or not $10^{2^n}+1$ is prime (or positive integers in general), I suggest using OpenPFGW. It has (among other things) an efficient implementation of a PRP test.

Using the ABC2 input format, we input this:

    ABC2 10^(2^$a)+1
    a: from 1 to 1000

and it outputs:

PFGW Version 3.6.6.64BIT.20120917.x86_Dev [GWNUM 27.8]


CPU Information (From Woltman v26 library code)
Intel(R) Core(TM) i7-2670QM CPU @ 2.20GHz
CPU speed: 2195.32 MHz, 4 hyperthreaded cores
CPU features: RDTSC, CMOV, Prefetch, MMX, SSE, SSE2, SSE4.1, SSE4.2
L1 cache size: 32 KB
L2 cache size: 256 KB, L3 cache size: 6 MB
L1 cache line size: 64 bytes
L2 cache line size: 64 bytes
TLBS: 64

Recognized ABC Sieve file:                                     
ABC2 File
10^(2^0)+1 is trivially prime!: 11                                    
10^(2^1)+1 is trivially prime!: 101                                    
Switching to Exponentiating using GMP                                    
Switching to Exponentiating using Woltman FFT's                                    
10^(2^13)+1 is composite: RES64: [64182BF8406B65C3] (2.4100s+0.0002s)
10^(2^14)+1 is composite: RES64: [C5FF6A4A68324D5A] (12.6942s+0.0003s)
10^(2^15)+1 is composite: RES64: [A874DC2BD3F1B9C8] (58.8378s+0.0003s)