Your question is still open. It is a special case of Schinzel's Hypothesis H applied to the polynomial $f(x)=x^{2^k}+1$.
As Bjorn mentions in his comment, the case of $k=1$ is a particularly famous unsolved problem. It is the fourth of Landau's problems (Edmund Landau was a famous German number theorist during the early twentieth century).
Your question is still open. It is a special case of Schinzel's Hypothesis H applied to the polynomial $f(x)=x^{2^k}+1$.