2,3

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

If (x0,y0) is the smallest positive solution to x^2 - D*y^2 = +-4, D = A003658(n), then a(n) = x_0/2 if D == 0 (mod 4), (x_0-y_0)/2 if D == 1 (mod 4). Note that y0 is given by A014046(n). - Jianing Song, Mar 23 2026

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

Jianing Song, Table of n, a(n) for n = 2..10000

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

Keith Matthews, Finding the fundamental unit of a real quadratic field

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]

(PARI) for(D=2, 200, if(isfundamental(D), print1(real(quadunit(D)), ", "))) \\ Jianing Song, Mar 22 2026

Cf. A014046, A003658, A014077, A003652.

Sequence in context: A145794 A176741 A266138 * A339612 A075257 A260618

Adjacent sequences: A013997 A013998 A013999 * A014001 A014002 A014003

Eric Rains (rains(AT)caltech.edu)

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

Offset corrected by Jianing Song, Mar 31 2019

approved