A067834 Numbers k such that in the ring Z[sqrt(3)] the norm of (-1+sqrt(3))^k-1 is prime.

2, 3, 7, 13, 19, 43, 61, 151, 257, 751, 859, 1453, 3767, 3889, 8171, 15959, 21499, 22679, 23297, 31277, 43609, 57037, 61961, 103087, 115931, 173647, 215959, 496073

Offset: 1

Author

Mike Oakes, Feb 09 2002

Keywords

• nonn
• nice
• more

Comments

The definition implies that k must be a prime.

The norm of (-1+sqrt(3))^k-1 is given by (-2)^k - lucasV(-2, -2, k)+1, where lucasV(-2, -2, k) is the solution of the recurrence relation v(n) = -2*v(n-1) + 2*v(n-2) with v(0)=2 and v(1)=-2.

a(29) > 517000. - Serge Batalov, Oct 24 2024

Examples

			3 is a term because (-2)^3-lucasV(-2,-2,3)+1 = -8-(-20)+1 = 13 and 13 is prime.
		

Links

• Mike Oakes, Posting to primenumbers list on Feb 08 2002
• Mike Oakes, 4 new forms of primes, digest of 4 messages in primenumbers Yahoo group, Feb 8 - Feb 11, 2002.

Crossrefs

Cf. A080040.

Programs

Mathematica

v[0] = 2; v[1] = -2; v[n_] := v[n] = -2*v[n-1] + 2*v[n-2] ; s = {}; Do[If[PrimeQ[(-2)^n - v[n] + 1], Print[n]; AppendTo[s, n]], {n, 8171}]; s (* Jean-François Alcover, Apr 18 2011 *)

PARI

isok(n)={ispseudoprime(([0, 1; 2, 2]^n*[2; 2])[1, 1] - 2^n - (-1)^n)} \\ Andrew Howroyd, Oct 24 2024

Extensions

Corrected and extended by Aurelien Gibier, Oct 24 2024

a(28) from Serge Batalov, Oct 24 2024