A341784 Norms of prime elements in Z[sqrt(-2)], the ring of integers of Q(sqrt(-2)).

2, 3, 11, 17, 19, 25, 41, 43, 49, 59, 67, 73, 83, 89, 97, 107, 113, 131, 137, 139, 163, 169, 179, 193, 211, 227, 233, 241, 251, 257, 281, 283, 307, 313, 331, 337, 347, 353, 379, 401, 409, 419, 433, 443, 449, 457, 467, 491, 499, 521, 523, 529, 547, 563

Offset: 1

Author

Jianing Song, Feb 19 2021

Keywords

• nonn
• easy

Comments

Also norms of prime ideals in Z[sqrt(-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 1, 2, 3 modulo 8 and the squares of primes congruent to 5, 7 modulo 8.

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

Examples

			norm(1 + sqrt(-2)) = norm(1 + sqrt(-2)) = 3;
norm(3 + sqrt(-2)) = norm(3 + sqrt(-2)) = 11;
norm(3 + 2*sqrt(-2)) = norm(3 + 2*sqrt(-2)) = 17;
norm(1 + 3*sqrt(-2)) = norm(1 + 3*sqrt(-2)) = 19.
		

Links

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

Crossrefs

Cf. A188510, A033203, A033200, A003628.

The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A002325.

The total number of elements with norm n is given by A033715.

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), this sequence (D=-8), A341785 (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

isA341784(n) = my(disc=-8); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)