A028877 Primes of form k^2 - 5.

11, 31, 59, 139, 191, 251, 479, 571, 1019, 1151, 1291, 1439, 1759, 1931, 2111, 2699, 3359, 4091, 5179, 5471, 6079, 6719, 8831, 10399, 12539, 13451, 14879, 17419, 20731, 23099, 26891, 27551, 28219, 30271, 30971, 33119, 33851, 34591, 35339, 39199, 41611, 44939, 49279

Offset: 1

Author

Patrick De Geest

Keywords

• nonn
• easy

Comments

These numbers are prime in Z but not in Z[sqrt(5)] nor in Z[phi] (where phi is the golden ratio), since (k - sqrt(5))(k + sqrt(5)) = ((k + 1) - 2*phi)((k - 1) + 2*phi) = k^2 - 5. - Alonso del Arte, Aug 27 2013

Examples

			31 is in the sequence as it is equal to 6^2 - 5.
59 is in the sequence since it is equal to 8^2 - 5.
95 is not in the sequence though it does equal 10^2 - 5.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..8000
• Patrick De Geest, Palindromic Quasipronics of the form n(n+x).
• Eric Weisstein's World of Mathematics, Near-Square Prime.

Crossrefs

Cf. A028875 (superset), A028876.

Programs

Magma

[a: n in [1..300] | IsPrime(a) where a is n^2-5]; // Vincenzo Librandi, Dec 01 2011

Mathematica

Select[Table[n^2 - 5, {n, 200}], PrimeQ] (* Harvey P. Dale, Jan 17 2011 *)

Formula

a(n) = A028875(A028876(n)). - Elmo R. Oliveira, Feb 22 2025