A003652 -id:A003652 - cp's OEIS frontend

A003658 Fundamental discriminants of real quadratic fields; indices of primitive positive Dirichlet L-series.

1, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40, 41, 44, 53, 56, 57, 60, 61, 65, 69, 73, 76, 77, 85, 88, 89, 92, 93, 97, 101, 104, 105, 109, 113, 120, 124, 129, 133, 136, 137, 140, 141, 145, 149, 152, 156, 157, 161, 165, 168, 172, 173, 177, 181, 184, 185, 188, 193, 197

Offset: 1

N. J. A. Sloane, Mira Bernstein, Eric W. Weisstein

All the prime numbers in the set of positive fundamental discriminants are Pythagorean primes (A002144). - Paul Muljadi, Mar 28 2008

Record numbers of prime divisors (with multiplicity) are 1, 5, and 4*A002110(n) for n > 0. - Charles R Greathouse IV, Jan 21 2022

Henri Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, pp. 515-519.
M. Pohst and Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, 1989, page 432.
Paulo Ribenboim, Algebraic Numbers, Wiley, NY, 1972, p. 97.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3001 from T. D. Noe)
Steven R. Finch, Class number theory, 2005. [Cached copy, with permission of the author]
Britta Habdank-Eichelsbacher, Unimodulare Gitter über Reell-Quadratischen Zahlkörpern, Ergänzungsreihe 95-005, Univ. Bielefeld, 1995. See Section 4.2.
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Fundamental Discriminant.
Eric Weisstein's World of Mathematics, Class Number.

Cf. A003652, A003657, A002144, A003646 (class numbers), A014000, A014046, A086669, A232931, A290098.

Union of A039955 and 4*A230375.

fundamentalDiscriminantQ[d_] := Module[{m, mod = Mod[d, 4]}, If[mod > 1, Return[False]]; If[mod == 1, Return[SquareFreeQ[d] && d != 1]]; m = d/4; Return[SquareFreeQ[m] && Mod[m, 4] > 1]; ]; Join[{1}, Select[Range[200], fundamentalDiscriminantQ]] (* Jean-François Alcover, Nov 02 2011, after Eric W. Weisstein *)
Select[Range[200], NumberFieldDiscriminant@Sqrt[#] == # &]  (* Alonso del Arte, Apr 02 2014, based on Arkadiusz Wesolowski's program for A094612 *)
max = 200; Drop[Select[Union[Table[Abs[MoebiusMu[n]] * n * 4^Boole[Not[Mod[n, 4] == 1]], {n, max}]], # < max &], 1] (* Alonso del Arte, Apr 02 2014 *)

v=[]; for(n=1,500,if(isfundamental(n),v=concat(v,n))); v

list(lim)=my(v=List()); forsquarefree(n=1,lim\4, listput(v, if(n[1]%4==1, n[1], 4*n[1]))); forsquarefree(n=lim\4+1, lim\1, if(n[1]%4==1, listput(v,n[1]))); Set(v) \\ Charles R Greathouse IV, Jan 21 2022

def is_fundamental(d):
    r = d % 4
    if r > 1 : return False
    if r == 1: return (d != 1) and is_squarefree(d)
    q = d // 4
    return is_squarefree(q) and (q % 4 > 1)
[1] + [n for n in (1..200) if is_fundamental(n)] # Peter Luschny, Oct 15 2018

Squarefree numbers (multiplied by 4 if not == 1 (mod 4)).

a(n) ~ (Pi^2/3)*n. There are (3/Pi^2)*x + O(sqrt(x)) terms up to x. - Charles R Greathouse IV, Jan 21 2022

More terms from Eric W. Weisstein and Jason Earls, Jun 19 2001

A003656 Discriminants of real quadratic fields with unique factorization.

Original entry on oeis.org

5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 41, 44, 53, 56, 57, 61, 69, 73, 76, 77, 88, 89, 92, 93, 97, 101, 109, 113, 124, 129, 133, 137, 141, 149, 152, 157, 161, 172, 173, 177, 181, 184, 188, 193, 197, 201, 209, 213, 217, 233, 236, 237, 241, 248, 249, 253, 268, 269

Offset: 1

N. J. A. Sloane, Mira Bernstein

Discriminants of real quadratic fields with class number 1.

Other than the term 8, every term is of one of the three following forms: (i) p, where p is a prime congruent to 1 modulo 4; (ii) 4p or 8p, where p is a prime congruent to 3 modulo 4; (iii) pq, where p, q are distinct primes congruent to 3 modulo 4. In fact, for a positive fundamental discriminant d, the class number of the real quadratic field of discriminant d is odd if and only if d = 8 or is of the form (i), (ii) or (iii). See Theorem 1 and Theorem 2 of Ezra Brown's link. - Jianing Song, Feb 24 2021

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
H. Cohen, Advanced Topics in Computational Number Theory, Springer, 2000, p. 534.
H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 576.
Pohst and Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, page 432.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
Ezra Brown, Class numbers of real quadratic number fields, Trans. Amer. Math. Soc. 190 (1974), 99-107.
Henri Cohen and X.-F. Roblot, Computing the Hilbert Class Field of Real Quadratic Fields, Math. Comp. 69 (2000), 1229-1244.
Eric Weisstein's World of Mathematics, Class Number
Index entries for sequences related to quadratic fields

Cf. A003652, A003658, A014602 (imaginary case).

For discriminants of real quadratic number fields with class number 2, 3, ..., 10, see A094619, A094612-A094614, A218156-A218160; see also A035120.

maxDisc = 269; t = Table[ {NumberFieldDiscriminant[ Sqrt[n] ], NumberFieldClassNumber[ Sqrt[n] ]}, {n, Select[ Range[2, maxDisc], SquareFreeQ] } ]; Union[ Select[ t, #[[2]] == 1 && #[[1]] <= maxDisc & ][[All, 1]]] (* Jean-François Alcover, Jan 24 2012 *)

is_fund_and_qfbcn_1 = lambda n: is_fundamental_discriminant(n) and QuadraticField(n, 'a').class_number() == 1
A003656 = lambda n: filter(is_fund_and_qfbcn_1, (1,2,..,n))
A003656(270) # Peter Luschny, Aug 10 2014

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 15 2002

A014000 First coordinate of fundamental unit of real quadratic field with discriminant A003658(n), n >= 2.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 5, 8, 2, 19, 5, 3, 27, 10, 3, 15, 131, 4, 17, 7, 11, 943, 170, 4, 4, 197, 447, 24, 13, 5035, 9, 5, 37, 118, 703, 11, 1520, 15371, 79, 35, 1595, 6, 87, 11, 28, 37, 25, 98, 10847, 6, 13, 3482, 6, 57731, 604, 24335, 63, 48, 1637147, 13, 478763

Offset: 2

Eric Rains (rains(AT)caltech.edu)

nonn

Taken from Cohen's table on pages 515-519. The table is indexed by the discriminant d = d(K) = A003658(n) of the real quadratic fields K. The fundamental unit is given as a pair of coordinates (a,b) = (A014000(n), A014046(n)) expressed in terms of the canonical integral basis (1,w) where w = (1+sqrt(d))/2 if d == 1 (mod 4), w = sqrt(d)/2 if d == 0 (mod 4).

The norm of this fundamental unit is A014077(n). The class number h(K) is A003652(n). - N. J. A. Sloane, Jun 14 2013

			Here is the start of Cohen's list of fundamental units: [0, 1], [1, 1], [2, 1], [1, 1], [3, 2], [2, 1], [5, 2], [8, 3], [2, 1], [19, 8], [5, 2], [3, 1], [27, 10], [10, 3], [3, 1], [15, 4], [131, 40],[4, 1], [17, 5], [7, 2], [11, 3], [943, 250], [170, 39], [4, 1], [4, 1], [197, 42], [447, 106], [24, 5], [13, 3], [5035, 1138], [9, 2], [5, 1], [37, 8], [118, 25], [703, 146], [11, 2], [1520, 273], [15371, 2968], [79, 15], [35, 6], [1595, 298], [6, 1], [87, 16], [11, 2], [28, 5], [37, 6], [25, 4], [98, 17], [10847, 1856], [6, 1], [13, 2], [3482, 531], [6, 1], [57731, 9384], [604, 97], [24335, 3588], [63, 10], [48, 7], [1637147, 253970], [13, 2], [478763, 72664], ... [_N. J. A. Sloane_, Jun 14 2013]

H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, pp. 515-519.

S. R. Finch, Class number theory [Cached copy, with permission of the author]
Keith Matthews, Finding the fundamental unit of a real quadratic field
Index entries for sequences related to quadratic fields

Cf. A014046, A003658, A014077, A003652.

Edited by N. J. A. Sloane, Jun 14 2013

Offset corrected by Jianing Song, Mar 31 2019

A014046 Second coordinate of fundamental unit of real quadratic field with discriminant A003658(n), n >= 2.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 1, 8, 2, 1, 10, 3, 1, 4, 40, 1, 5, 2, 3, 250, 39, 1, 1, 42, 106, 5, 3, 1138, 2, 1, 8, 25, 146, 2, 273, 2968, 15, 6, 298, 1, 16, 2, 5, 6, 4, 17, 1856, 1, 2, 531, 1, 9384, 97, 3588, 10, 7, 253970, 2, 72664, 7, 3, 6440, 5, 521904, 12, 1, 1, 13

Offset: 2

Eric Rains (rains(AT)caltech.edu)

nonn

See A014000 for much more about this sequence. - N. J. A. Sloane, Jun 14 2013

H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, pp. 515-519.

S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]

Cf. A014000, A003658, A014077, A003652.

lista(nn) = { for (n=2, nn, if (isfundamental(n), nc = core(n); m = Mod (nc, 4); if ((m == 2) || (m == 3), d = 1); if ((m == 1), d = 4); b = 1; a = 0; while (a == 0, v = nc*b^2; if (issquare(v-d), a = sqrtint(v-d), if (issquare(v+d), a = sqrtint(v+d))); if (a == 0, b++; );); print1(b, ", ");););} \\ Michel Marcus, Sep 25 2018

Offset corrected by Jianing Song, Mar 31 2019

A014077 Norm of fundamental unit of real quadratic field with discriminant A003658(n), n >= 2.

Original entry on oeis.org

-1, -1, 1, -1, -1, 1, 1, 1, -1, 1, -1, -1, -1, 1, -1, 1, 1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, 1, 1, -1, -1, -1, 1, -1, -1, 1, 1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -1

Offset: 2

Eric Rains (rains(AT)caltech.edu)

sign

See A014000 for details about the indexing.

H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, pp. 515-519.

Cf. A003658, A003652, A014000, A014046.

Offset corrected by Jianing Song, Mar 31 2019

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A003658 Fundamental discriminants of real quadratic fields; indices of primitive positive Dirichlet L-series.

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A003656 Discriminants of real quadratic fields with unique factorization.

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A014000 First coordinate of fundamental unit of real quadratic field with discriminant A003658(n), n >= 2.

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A014046 Second coordinate of fundamental unit of real quadratic field with discriminant A003658(n), n >= 2.

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A014077 Norm of fundamental unit of real quadratic field with discriminant A003658(n), n >= 2.

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