Near-square prime

Definition

A near-square prime of type $n^{2}-k$ for fixed $k$ is a prime that can be written in the form $n^{2}-k$. Note that we consider near-square primes for $k$ not a perfect square, because if $k$ is a perfect square, then $n^{2}-k$ factorizes algebraically.

According to the Bunyakovsky conjecture, for any fixed $k$ not a perfect square, there should be infinitely many near-square primes of the form $n^{2}-k$.

Occurrence

Value of $k$	Polynomial $n^{2}-k$	First few prime values
-1	$n^{2}+1$	2, 5, 17, 37, 101, 197, 257, [SHOW MORE] 401, 577, 677, 1297, 1601, 2917, 3137, 4357, 5477, 7057, 8101, 8837, 12101, 13457, 14401, 15377, 15877, 16901, 17957, 21317, 22501, 24337, 25601, 28901, 30977, 32401, 33857, 41617, 42437, 44101, 50177 View list on OEIS