A239842 Numbers n such that the Eisenstein integer ((1-ω)^n+1)/(2-ω) has prime norm, where ω = - 1/2 + sqrt(-3)/2.

5, 11, 31, 37, 47, 53, 97, 163, 167, 509, 877, 1061, 2027, 2293, 3011, 6803, 8423, 13627, 20047, 28411, 50221, 50993, 71453, 152809, 272141, 505823, 1353449

Offset: 1

Author

Serge Batalov, Mar 27 2014

Keywords

• more
• nonn
• hard

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.

Examples

			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.
		

Links

• Henri Lifchitz & Renaud Lifchitz: PRP Records. Probable Primes Top 10000.

Crossrefs

Cf. A125743 = Primes p such that (3^p - 3^((p+1)/2) + 1)/7 is prime.

Cf. A125744 = Primes p such that (3^p + 3^((p+1)/2) + 1)/7 is prime.

Cf. A066408 = Numbers n such that the Eisenstein integer has prime norm.

Cf. A124112 = Numbers n such that ((1+I)^n+1)/(2+I) is a Gaussian prime.

Programs

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 */