cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

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Author

Eric Rains (rains(AT)caltech.edu)

Keywords

Comments

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

Examples

			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]

References

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

Links

Crossrefs

Cf. A014046, A003658, A014077, A003652.

Extensions

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

Views

Author

Eric Rains (rains(AT)caltech.edu)

Keywords

Comments

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

References

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

Links

Crossrefs

Cf. A014000, A003658, A014077, A003652.

Programs

  • PARI
    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

Extensions

Offset corrected by Jianing Song, Mar 31 2019

A306638 a(n) is the norm of the fundamental unit of binary quadratic forms with discriminant D = A079896(n).

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, 1, -1, -1, -1
Offset: 1

Views

Author

Jianing Song, Mar 02 2019

Keywords

Comments

The fundamental unit of binary quadratic forms with discriminant D is the number (x_1 + (y_1)*sqrt(D))/2, where (x_1,y_1) is the smallest solution to x^2 - D*y^2 = +-4. Each term is either -1 or 1 depending on whether (x_1)^2 - D*(y_1)^2 = -4 or 4.
All solutions to x^2 - D*y^2 = +-4 are given by the identity (x_n + (y_n)*sqrt(D))/2 = ((x_1 + (y_1)*sqrt(D))/2)^n.
The discriminants D corresponding to (x_1)^2 - D*(y_1)^2 = -4 are listed in A226696.

Examples

			Fundamental units and their norms for the first 15 discriminants in the form (X + Y*sqrt(D))/2 (N = (X^2 - D*Y^2)/4 are the corresponding norms) are:
   D |  X |  Y |  N
   5 |  1 |  1 | -1
   8 |  2 |  1 | -1
  12 |  4 |  1 |  1
  13 |  3 |  1 | -1
  17 |  8 |  2 | -1
  20 |  4 |  1 | -1
  21 |  5 |  1 |  1
  24 | 10 |  2 |  1
  28 | 16 |  3 |  1
  29 |  5 |  1 | -1
  32 |  6 |  1 |  1
  33 | 46 |  8 |  1
  37 | 12 |  2 | -1
  40 |  6 |  1 | -1
  41 | 64 | 10 | -1

References

  • D. A. Buell, Binary Quadratic Forms, Springer, 1989, Sections 3.2 and 3.3, pp. 31-48.

Links

Crossrefs

Cf. A079896, A226696.
A014077 is a subsequence listing the corresponding values for only fundamental discriminants (A003658).

Programs

  • Julia
    using Nemo
function b(D)
    for j in 1:10000
        issquare(D*j^2 - 4) && return -1
        issquare(D*j^2 + 4) && return 1
    end
0 end
F = findall(n -> ZZ(n) % 4 <= 1 && !issquare(ZZ(n)), 1:100)
map(n -> b(ZZ(n)), F) |> println # Peter Luschny, Mar 08 2019
  • PARI
    b(D) = for(n=1, oo, if(issquare(D*n^2-4), return(-1)); if(issquare(D*n^2+4), return(1)))
for(n=2, 200, if(n%4 <= 1 && !issquare(n), print1(b(n), ", ")))

Formula

a(n) = -1 if D = A079896(n) is in A226696, otherwise 1.

Extensions

Offset changed to 1 by Robin Visser, Jun 09 2025
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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