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Discriminant of quadratic field Q(sqrt(n)).
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%I #41 May 16 2024 20:35:40

%S 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,

%T 12,28,29,120,124,8,33,136,140,1,37,152,156,40,41,168,172,44,5,184,

%U 188,12,1,8,204,13,53,24,220,56,57,232,236,60,61,248,28,1,65,264,268,17,69

%N Discriminant of quadratic field Q(sqrt(n)).

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

%C 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

%H T. D. Noe, <a href="/A037449/b037449.txt">Table of n, a(n) for n = 1..1000</a>

%H S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Class number theory</a>

%H Steven R. Finch, <a href="/A000924/a000924.pdf">Class number theory</a> [Cached copy, with permission of the author]

%F 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

%t Table[NumberFieldDiscriminant[Sqrt[n]], {n, 100}] (* _Artur Jasinski_, Jan 27 2012 *)

%o (PARI) vector(150,n,quaddisc(n))

%o (Sage)

%o [fundamental_discriminant(n) for n in (1..69)] # _Peter Luschny_, Oct 15 2018

%Y Cf. A204993, A007913.

%K nonn,easy

%O 1,2

%A _Jason Earls_, Jun 30 2001