A391271 Negated discriminants of imaginary quadratic fields F such that 16 is an 8th power in F_2, where F_2 is the completion of F with respect to a place above 2.

4, 8, 56, 68, 120, 132, 136, 164, 184, 228, 248, 260, 264, 292, 312, 328, 356, 376, 388, 420, 440, 452, 456, 516, 520, 548, 568, 580, 584, 632, 644, 696, 708, 712, 740, 760, 772, 776, 804, 824, 836, 840, 868, 888, 904, 932, 952, 964, 996, 1016, 1028, 1032, 1060, 1092, 1096, 1124, 1144, 1160

Offset

1,1

Comments

Let F be a number field. The Grunwald-Wang theorem states that:

- if t is a q-th power in every completion of F for an odd prime power q, then t is a q-th power in F;

- if t is a (2^n)-th power in every completion of F, then t is either a (2^n)-th power in F or 2^2^(n-1) times a (2^n)-th power (if n >= 3).

As a result, by the lemma below, F satisfies the local-global principle for n-th powers if and only if

- either at least one of -1, -2, and 2 is a square in F;

- or none of -1, -2, or 2 has a square root in F_v, where v is a place above 2.

In general there can be different places above p. But if F/Q is a Galois extension (for example if F is a quadratic extension), then the Galois group acts transitively on the prime ideals above p, and so all F_v are isomorphic.

Lemma. Let F be a field of characteristic != 2, and n >= 3. Then X^2^n - 2^2^(n-1) has a root in F if and only if at least one of -1, -2, and 2 is a square in F.

Proof. The "if" part is clear since (+-1+-i)^8 = (+-sqrt(-2))^8 = (+-sqrt(2))^8 = 16, so (1+i)^2^n = (sqrt(-2))^2^n = (sqrt(2))^2^n = 2^2^(n-1). Conversely, note that X^2^n - 2^2^(n-1) = (X^2+2*X+2)*(X^2-2*X+2)*(X^2+2)*(X^2-2)*Product_{k=3..n-1} (X^2^k + 2^2^(k-1)). If x is a root to X^2^k + 2^2^(k-1) = 0 (k >= 3), then x/(1 + x^2^(k-1)/2^2^(k-2)) is a root to X^2^k + 1 = 0, and so F0(zeta_{2^{k+1}}) is included in F, where F0 is the prime field of F. But F0(zeta_8) contains the square roots of -1, -2, and 2.

Links

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

Ishan Levy, Local-global principle for n-th powers.

Wikipedia, Grunwald-Wang theorem.

Formula

If d < 0 is a fundamental discriminant, then -d is a term if and only if d == 8, 28, 56, 60 (mod 64).

Example

Let d be the fundamental discriminant of a quadratic field F. Then:
- if d == 1 (mod 8), then F_2 = Q_2, in which 16 is not an 8th power;
- if d == 5 (mod 8), then F_2 = Q_2(sqrt(-3)), in which 16 is not an 8th power;
- if d == 12 (mod 32), then F_2 = Q_2(sqrt(3)), in which 16 is not an 8th power;
- if d == 28 (mod 32), then F_2 = Q_2(sqrt(-1)), in which 16 is an 8th power;
- if d == 8 (mod 64), then F_2 = Q_2(sqrt(2)), in which 16 is an 8th power;
- if d == 24 (mod 64), then F_2 = Q_2(sqrt(6)), in which 16 is not an 8th power;
- if d == 40 (mod 64), then F_2 = Q_2(sqrt(-6)), in which 16 is not an 8th power;
- if d == 56 (mod 64), then F_2 = Q_2(sqrt(-2)), in which 16 is an 8th power.

Prog

(PARI) isA391271(n) = isfundamental(-n) && ((-n)%32==28 || (-n)%64==8 || (-n)%64==56)

Crossrefs

Cf. A391409 (d > 0 case), A003657 (negated fundamental discriminants of imaginary quadratic fields).

Sequence in context: A347511 A123106 A123288 * A192200 A063083 A349859

Adjacent sequences: A391268 A391269 A391270 * A391272 A391273 A391274

Keyword

nonn,easy

Author

Jianing Song, Jan 12 2026

Status

approved