A near-square prime is a prime number $p$ of the form $n^2 + k$ , with $n$ being any integer and $0 < |k| < |n|$ also an integer. Since for any nonzero real number $x$ it is always the case that $x^2 \geq 0$ , 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.