A323697 Primes p such that the norm of the quadratic-field analog of Mersenne numbers M_{p,alpha} = (alpha^p - 1)/(alpha - 1), with alpha = 2 + sqrt(2), is a rational prime.

2, 3, 5, 7, 11, 73, 89, 233, 4057, 8429, 12451, 21191

Offset

1,1

Comments

The corresponding norms of the Mersenne primes are in A232993. The values corresponding to a(9)-a(12) are about 3.633...*10^2163, 1.279...*10^4495, 1.005...*10^6640, and 9.180...*10^11300.

Links

Table of n, a(n) for n=1..12.

Sushma Palimar and B. R. Shankar, Mersenne Primes in Real Quadratic Fields, Journal of Integer Sequences, Vol. 15 (2012), #12.5.6.

Example

5 is in the sequence since M_{5,alpha} = (alpha^5 - 1)/(alpha - 1) = 97 + 67*sqrt(2), and its norm is 97^2 - 2*67^2 = 431 is prime.

Mathematica

norM[p_] := Module[{a = (Sqrt[2] + 2)}, CoefficientList[MinimalPolynomial[(a^p - 1)/(a - 1), Sqrt[2]], Sqrt[2]][[1]] - 2]; Select[Prime /@ Range[100], PrimeQ[norM[#]] &]

Crossrefs

Cf. A000043, A000668, A232993.

Sequence in context: A070892 A075800 A067909 * A103850 A288369 A226732

Adjacent sequences: A323694 A323695 A323696 * A323698 A323699 A323700

Keyword

nonn,more

Author

Amiram Eldar, Jan 24 2019

Status

approved