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A014000 First coordinate of fundamental unit of real quadratic field with discriminant A003658(n), n >= 2. 5

%I #42 Dec 14 2019 01:46:42

%S 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,

%T 447,24,13,5035,9,5,37,118,703,11,1520,15371,79,35,1595,6,87,11,28,37,

%U 25,98,10847,6,13,3482,6,57731,604,24335,63,48,1637147,13,478763

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

%C 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).

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

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

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

%H Keith Matthews, <a href="http://www.numbertheory.org/php/unit.html">Finding the fundamental unit of a real quadratic field</a>

%H <a href="/index/Qua#quadfield">Index entries for sequences related to quadratic fields</a>

%e 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]

%Y Cf. A014046, A003658, A014077, A003652.

%K nonn

%O 2,3

%A Eric Rains (rains(AT)caltech.edu)

%E Edited by _N. J. A. Sloane_, Jun 14 2013

%E Offset corrected by _Jianing Song_, Mar 31 2019

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