Dear All, I wonder what the numbers(primes) of the form of 2^n + x^2 (where n is even) are called? What are their properties? Any references to look at? Thank you.

Primes of the form $2^n+k$ have been considered, see the talk of Carl Pomerance. Among the square numbers $k=y^2$ the case $k=1$ is the most famous one, e.g., primes of the from $2^n+1$. Then necessarily $n$ is a power of $2$, so that these primes are just the Fermat primes $F_k=2^{2^k}+1$. Another special case are the primes of the form $2^n+n^2$, see sequence A064539 at integer sequences. Then necessarily $n\equiv 0 \mod 3$. 

