A037449 Discriminant of quadratic field Q(sqrt(n)).

1, 8, 12, 1, 5, 24, 28, 8, 1, 40, 44, 12, 13, 56, 60, 1, 17, 8, 76, 5, 21, 88, 92, 24, 1, 104, 12, 28, 29, 120, 124, 8, 33, 136, 140, 1, 37, 152, 156, 40, 41, 168, 172, 44, 5, 184, 188, 12, 1, 8, 204, 13, 53, 24, 220, 56, 57, 232, 236, 60, 61, 248, 28, 1, 65, 264, 268, 17, 69

Offset: 1

Author

Jason Earls, Jun 30 2001

Keywords

• nonn
• easy

Comments

For the discriminant of the quadratic field Q(sqrt(-n)), see A204993.

a(n) is the smallest positive N such that (n/k) = (n/(k mod N)) for every odd k that is coprime to n, where (n/k) is the Jacobi symbol. As we have Dirichlet's theorem on arithmetic progressions, a(n) is also the smallest positive N such that (n/p) = (n/(p mod N)) for every odd prime p that is not a factor of n. - Jianing Song, May 16 2024

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• S. R. Finch, Class number theory
• Steven R. Finch, Class number theory [Cached copy, with permission of the author]

Crossrefs

Cf. A204993, A007913.

Programs

Mathematica

Table[NumberFieldDiscriminant[Sqrt[n]], {n, 100}] (* Artur Jasinski, Jan 27 2012 *)

PARI

vector(150,n,quaddisc(n))

Sage

[fundamental_discriminant(n) for n in (1..69)] # Peter Luschny, Oct 15 2018

Formula

Let b(n) = A007913(n), then a(n) = b(n) if b(n) == 1 (mod 4) and 4*b(n) otherwise. - Jianing Song, May 16 2024