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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.
0
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
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
KEYWORD
nonn,more
AUTHOR
Amiram Eldar, Jan 24 2019
STATUS
approved