Prime numbers of a special kind Is it proved that there is only a finite number of prime numbers of the form $2^{2n} + 2^n + 1$ (where $n \in \mathbb{N}$)? 
 A: No, it is not proved that there is only a finite number of primes of the form $2^{2n} + 2^n + 1$. 
As mentioned by Gerry Myerson in a comment it is however known that an $n$ such that the number of the above form is prime needs to be a power of $3$. 
Consequently, standard heuristics suggest that there are only finitely many prime numbers of this form. For given $k$, the naive expectation says that the probability that 
$$
2^{2\ 3^k } + 2^{3^k} + 1
$$
is prime is about the reciprocal of the natural logarithm of this number, 
that is it is less than $3^{-k}$ and the series $\sum_{k} 3^{-k}$ being convergent one only expects a finite number of primes of this form in total. 
Note, however, that if one did not have or know the strong restriction that $n$ needs to be a power of $3$, the expectation would, or at least could depending on the precise restriction one has, be different (compare this situation to the fact that one expects only finitely many Fermat numbers to be prime yet infinitely many Mersenne primes).
It seems the only known primes of this form are $7, 73, 262657$ corresponding to $k=0,1,2$ the next, that is $k=3$, is $18014398643699713 = 2559 \times 71119 \times 9768539$ and thus composite and it is known for a few subsequent ones that they are also composite (see the comment of joro).  
It would not be surprsing if the only primes of this form were the three known ones, but it would also not be surprising if there were a few more. It would only be surprising if there were many (in particular if there were infinitely many) primes among numbers of this form. 
This like most questions of this general type (primes in sparse sequences) is however an open problem and will likely remain so for quite some time.  
For completeness I repeat the reference provided by Gerry Myerson http://oeis.org/A051154.
