near-square prime

A near-square prime is a prime number $p$ of the form $n^2+k$, with $n$ being any integer and also an integer. Since for any nonzero real number $x$ it is always the case that $x^2\ge 0$, it doesn’t matter if $n$ is negative.

Fermat primes are near-square primes for $k=1$ with the additional requirement that $n=2^{2^m-1}$, while Carol primes are near-square primes for $k=-2$ with the additional requirement that $n=2^m-1$.

For $k=-1$, only $n=2$ gives a prime, namely 3.