A341785 Norms of prime elements in Z[(1+sqrt(-11))/2], the ring of integers of Q(sqrt(-11)).

3, 4, 5, 11, 23, 31, 37, 47, 49, 53, 59, 67, 71, 89, 97, 103, 113, 137, 157, 163, 169, 179, 181, 191, 199, 223, 229, 251, 257, 269, 289, 311, 313, 317, 331, 353, 361, 367, 379, 383, 389, 397, 401, 419, 421, 433, 443, 449, 463, 467, 487, 499, 509, 521

Offset: 1

Author

Jianing Song, Feb 19 2021

Keywords

• nonn
• easy

Comments

Also norms of prime ideals in Z[(1+sqrt(-11))/2], which is a unique factorization domain. The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.

Consists of the primes congruent to 0, 1, 3, 4, 5, 9 modulo 11 and the squares of primes congruent to 2, 6, 7, 8, 10 modulo 5.

For primes p == 1, 3, 4, 5, 9 (mod 11), there are two distinct ideals with norm p in Z[(1+sqrt(-11))/2], namely (x + y*(1+sqrt(-11))/2) and (x + y*(1-sqrt(-11))/2), where (x,y) is a solution to x^2 + x*y + 3*y^2 = p; for p = 11, (sqrt(-11)) is the unique ideal with norm p; for p == 2, 6, 7, 8, 10 (mod 11), (p) is the only ideal with norm p^2.

Examples

			norm((1 + sqrt(-11))/2) = norm((1 - sqrt(-11))/2) = 3;
norm((3 + sqrt(-11))/2) = norm((3 - sqrt(-11))/2) = 5;
norm((9 + sqrt(-11))/2) = norm((9 - sqrt(-11))/2) = 23;
norm((5 + 3*sqrt(-11))/2) = norm((5 - 3*sqrt(-11))/2) = 31.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A011582, A056874, A296920, A191060.

The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A035179.

The total number of elements with norm n is given by A028609.

Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), this sequence (D=-11), A341786 (D=-15*), A341787 (D=-19), A091727 (D=-20*), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

Programs

PARI

isA341785(n) = my(disc=-11); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)