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.

5												149
4					29		53
3								67		103
2			11						83
1		5		17		37				101
0	1	4	9	16	25	36	49	64	81	100	121	144
$-1$		3
$-2$			7		23		47		79
$-3$										97
$-4$
$-5$						31		59				139

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.

Title	near-square prime
Canonical name	NearsquarePrime
Date of creation	2013-03-22 18:57:37
Last modified on	2013-03-22 18:57:37
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	6
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11A41