Call a number of the form n^2 - k a "near-square number." Numbers of the form n^2 - 5 for n = 1, 2, ... are -4, -1, 4, 11, 20, 31, 44, 59, 76, 95, ... (OEIS A028875). These are prime for indices n = 4, 6, 8, 12, 14, 16, ... (OEIS A028876), corresponding to the primes 11, 31, 59, 139, 191, 251, 479, ... (OEIS A028877). Numbers of the form n^2 - 4 for n = 1, 2, ... are -3, 0, 5, 12, 21, 32, 45, 60, ... (OEIS A028347). Since they can always be factored as (n - 2)(n + 2) for n>2, the only (positive) prime of this form is 5.
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