A391366 Norms of prime ideals of Z[sqrt(-6)].

2, 3, 5, 7, 11, 29, 31, 53, 59, 73, 79, 83, 97, 101, 103, 107, 127, 131, 149, 151, 169, 173, 179, 193, 197, 199, 223, 227, 241, 251, 269, 271, 289, 293, 313, 317, 337, 347, 361, 367, 389, 409, 419, 433, 439, 443, 457, 461, 463, 467, 487, 491, 509, 529, 557, 563, 577, 587, 601, 607, 631, 653, 659, 673, 677, 683

Offset

1,1

Comments

Consists of 2, 3, the primes p == 1, 5, 7, 11 (mod 24) and the squares of primes p such that p == 13, 17, 19, 23 (mod 24).

Note that Q(sqrt(-6)) has class number 2.

For primes p == 1, 7 (mod 24), there are two distinct ideals with norm p in Z[sqrt(-6)], namely (x + y*sqrt(-6)) and (x - y*sqrt(-6)), where (x,y) is a solution to x^2 + 6*y^2 = p.

For p == 5, 11 (mod 24), there are also two distinct ideals with norm p, namely (p, x+y*sqrt(-6)) and (p, x-y*sqrt(-6)), where (x,y) is a solution to x^2 + 6*y^2 = p^2 with y != 0; (2, sqrt(-6)) and (3, sqrt(-6)) are respectively the unique ideal with norm 2 and 3.

For p == 13, 17, 19, 23 (mod 24), (p) is the only ideal with norm p^2.

Links

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

Example

|(5, 1 +- 2*sqrt(-6))| = 5, |(1 +- sqrt(-6))| = 7, |(11, 5 +- 4*sqrt(-6))| = 11, |(29, 25 +- 3*sqrt(-6))| = 29, |(5 +- sqrt(-6))| = 31, ...

Mathematica

seq[lim_] := Sort[Join[{2, 3}, Select[Range[lim], MemberQ[{1, 5, 7, 11}, Mod[#, 24]] && PrimeQ[#] &], Select[Range[Sqrt[lim]], MemberQ[{13, 17, 19, 23}, Mod[#, 24]] && PrimeQ[#] &]^2]]; seq[700] (* Amiram Eldar, Mar 30 2026 *)

Prog

(PARI) isA391366(n, {disc=-24}) = (isprime(n) && kronecker(disc, n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc, n)==-1)

Crossrefs

Cf. A109017 ({kronecker(-24,n)}), whose inverse Moebius transform A192013 gives the numbers of distinct ideals with each norm (i.e., the coefficients of Dedekind zeta function).

Cf. A296924 (primes not inert in Q(sqrt(-6))), A157437 (primes decomposing), A191059 (prime remaining inert).

Cf. A033199 ((p) is the product of two principal ideals), A084865 ((p) is the product of two non-principal ideals).

Norms of prime ideals in the ring of integers of quadratic fields of class number 1: A391371 (D=24), A391370 (D=21), A391369 (D=12), A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), A341785 (D=-11), A341787 (D=-19), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163).

Norms of prime ideals in the ring of integers of quadratic fields of class number 2: A391367 (D=40), A341786 (D=-15), A091727 (D=-20), this sequence (D=-24).

Sequence in context: A387593 A167119 A165682 * A296924 A343423 A175711

Adjacent sequences: A391363 A391364 A391365 * A391367 A391368 A391369

Keyword

nonn,easy

Author

Jianing Song, Dec 07 2025

Status

approved