cp's OEIS Frontend

A348725 Decimal expansion of the absolute value of one of the negative roots of Shanks' simplest cubic associated with the prime p = 19.

%I A348725 #15 Nov 08 2021 16:32:34
%S A348725 1,2,8,5,1,4,2,4,8,1,8,2,9,7,8,5,3,6,4,3,9,4,1,1,9,8,7,3,5,3,0,6,2,7,
%T A348725 4,1,3,4,2,6,7,8,0,9,2,5,7,2,2,6,1,6,9,4,1,5,2,5,6,6,7,0,6,9,8,6,1,9,
%U A348725 9,1,7,2,1,9,7,9,5,2,3,0,5,0,7,0,3,8,0,4,2,3,8,9,7,4,2,9,8,7,3,9
%N A348725 Decimal expansion of the absolute value of one of the negative roots of Shanks' simplest cubic associated with the prime p = 19.
%C A348725 Let a be an integer and let p be a prime of the form a^2 + 3*a + 9 (see A005471). Shanks introduced a family of cyclic cubic fields generated by the roots of the polynomial x^3 - a*x^2 - (a + 3)*x - 1.
%C A348725 In the case a = 2, corresponding to the prime p = 19, Shanks' cyclic cubic is x^3 - 2*x^2 - 5*x - 1 of discriminant 19^2. The polynomial has three real roots, one positive and two negative. Let r_0 = 3.507018644... denote the positive root. The other roots are r_1 = - 1/(1 + r_0) = - 0.2218761622... and r_2 = - 1/(1 + r_1) = - 1.2851424818.... See A348723 (r_0) and A348724 (|r_1|).
%C A348725 Here we consider the absolute value of the root r_2.
%H A348725 T. W. Cusick and Lowell Schoenfeld, <a href="https://doi.org/10.1090/S0025-5718-1987-0866105-8">A table of fundamental pairs of units in totally real cubic fields</a>, Math. Comp. 48 (1987), 147-158 (see case 9 in the table)
%H A348725 D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1974-0352049-8">The simplest cubic fields</a>, Math. Comp., 28 (1974), 1137-1152
%F A348725 |r_2| = sin(Pi/19)*sin(7*Pi/19)*sin(8*Pi/19)/(sin(2*Pi/19)*sin(3*Pi/19)* sin(5*Pi/19)) = 1/(8*cos(Pi/19)*cos(7*Pi/19)*cos(8*Pi/19)).
%F A348725 |r_2| = Product_{n >= 0} (19*n+1)*(19*n+7)*(19*n+8)*(19*n+11)*(19*n+12)*(19*n+18)/ ( (19*n+2)*(19*n+3)*(19*n+5)*(19*n+14)*(19*n+16)*(19*n+17) ).
%F A348725 |r_2| = 2*(cos(Pi/19) + cos(7*Pi/19) - cos(8*Pi/19)) - 1.
%F A348725 Let z = exp(2*Pi*i/19). Then
%F A348725 |r_2| = abs( (1 - z)*(1 - z^7)*(1 - z^8)/((1 - z^2)*(1 - z^3)*(1 - z^5)) ).
%F A348725 Note: C = {1, 7, 8, 11, 12, 18} is the subgroup of nonzero cubic residues in the finite field Z_19 with cosets 2*C = {2, 3, 5, 14, 16, 17} and 4*C = {4, 6, 9, 10, 13, 15}.
%F A348725 Equals -1 + (-1)^(1/19) + (-1)^(7/19) - (-1)^(8/19) + (-1)^(11/19) - (-1)^(12/19) - (-1)^(18/19). - _Peter Luschny_, Nov 08 2021
%e A348725 1.28514248182978536439411987353062741342678092572261 ...
%p A348725 evalf(sin(Pi/19)*sin(7*Pi/19)*sin(8*Pi/19)/(sin(2*Pi/19)*sin(3*Pi/19)*sin(5*Pi/19)), 100);
%t A348725 RealDigits[Sin[Pi/19]*Sin[7*Pi/19]*Sin[8*Pi/19]/(Sin[2*Pi/19]*Sin[3*Pi/19]*Sin[5*Pi/19]), 10, 100][[1]] (* _Amiram Eldar_, Nov 08 2021 *)
%Y A348725 Cf. A005471, A160389, A255240, A255241, A255249, A348720 - A348729.
%K A348725 nonn,cons,easy
%O A348725 1,2
%A A348725 _Peter Bala_, Oct 31 2021