OFFSET
1,1
COMMENTS
These numbers are sometimes called Eisenstein-Mersenne cofactors EQ(n).
The p-th Eisenstein-Mersenne cofactor can be written as EQ(p) = (3^p + Legendre(3, p) * 3^((p + 1)/2) + 1)/7.
Following an idea of Harsh Aggarwal, some of these numbers have been discovered as by-products of the search for prime Eisenstein-Mersenne norms. The reason of that is the Aurifeuillan factorization of T(k) = 3^(3k) + 1 with k odd. These numbers can be written as T(k) = (3^k + 1)*EM(k)*EQ(k)*7, EM(k) is the norm of the Eisenstein-Mersenne (1-ω)^k-1, while EQ(k) is the norm of ((1-ω)^a[n]+1)/(2-ω).
These numbers have been proved prime only up to exponent a(19) = 20047.
Next term a(28) > 1500000.
LINKS
Henri Lifchitz & Renaud Lifchitz: PRP Records. Probable Primes Top 10000.
EXAMPLE
For n = 3: ((1-ω)^31+1)/(2-ω) is an Eisenstein prime because its norm, (3^31-3^16+1)/7 = 88239050462461, is prime.
PROG
(PARI) forprime(n=3, 2300, if(ispseudoprime((3^n+kronecker(3, n)*3^((n+1)/2)+1)/7), print1(n ", "))); /* Serge Batalov, Mar 29 2014 */
CROSSREFS
KEYWORD
more,nonn,hard
AUTHOR
Serge Batalov, Mar 27 2014
STATUS
approved