cp's OEIS Frontend

A348724 Decimal expansion of the absolute value of one of the negative roots of Shanks' simplest cubic associated with the prime p = 19.

%I A348724 #16 Nov 08 2021 16:32:26
%S A348724 2,2,1,8,7,6,1,6,2,2,6,3,1,9,0,9,3,4,2,6,6,6,8,0,0,5,0,1,8,5,0,5,0,6,
%T A348724 1,5,5,9,9,1,9,5,4,9,4,4,0,7,7,5,2,7,3,3,6,0,0,9,1,5,1,0,8,4,9,0,9,8,
%U A348724 5,2,4,2,8,4,1,4,9,6,9,2,0,8,7,2,1,9,9,1,6,9,6,4,5,1,1,0,3,3,2,2
%N A348724 Decimal expansion of the absolute value of one of the negative roots of Shanks' simplest cubic associated with the prime p = 19.
%C A348724 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 A348724 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 and A348725.
%C A348724 Here we consider the absolute value of the root r_1. In Cusick and Schoenfeld r_1 is denoted by E_2. See case 9 in the table.
%H A348724 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
%H A348724 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 A348724 |r_1| = 2*(cos(3*Pi/19) + cos(5*Pi/19) - cos(2*Pi/19)) - 1.
%F A348724 |r_1| = sin(2*Pi/19)*sin(3*Pi/19)*sin(5*Pi/19)/(sin(4*Pi/19)*sin(6*Pi/19) *sin(9*Pi/19)) = 1/(8*cos(2*Pi/19)*cos(3*Pi/19)*cos(5*Pi/19)).
%F A348724 |r_1| = Product_{n >= 0} (19*n+2)*(19*n+3)*(19*n+5)*(19*n+14)*(19*n+16)*(19*n+17)/( (19*n+4)*(19*n+6)*(19*n+9)*(19*n+10)*(19*n+13)*(19*n+15) ).
%F A348724 Let z = exp(2*Pi*i/19). Then
%F A348724 |r_1| = abs( (1 - z^2)*(1 - z^3)*(1 - z^5)/((1 - z^4)*(1 - z^6)*(1 - z^9)) ).
%F A348724 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 A348724 Equals -1 - (-1)^(2/19) + (-1)^(3/19) + (-1)^(5/19) - (-1)^(14/19) - (-1)^(16/19) + (-1)^(17/19). - _Peter Luschny_, Nov 08 2021
%e A348724 0.22187616226319093426668005018505061559919549440775...
%p A348724 evalf(sin(2*Pi/19)*sin(3*Pi/19)*sin(5*Pi/19)/(sin(4*Pi/19)*sin(6*Pi/19)*sin(9*Pi/19)), 100);
%t A348724 RealDigits[Sin[2*Pi/19]*Sin[3*Pi/19]*Sin[5*Pi/19]/(Sin[4*Pi/19]*Sin[6*Pi/19]*Sin[9*Pi/19]), 10, 100][[1]] (* _Amiram Eldar_, Nov 08 2021 *)
%o A348724 (PARI) sin(2*Pi/19)*sin(3*Pi/19)*sin(5*Pi/19)/(sin(4*Pi/19)*sin(6*Pi/19)*sin(9*Pi/19)) \\ _Michel Marcus_, Nov 08 2021
%Y A348724 Cf. A005471, A160389, A255240, A255241, A255249, A348720 - A348729.
%K A348724 nonn,cons,easy
%O A348724 0,1
%A A348724 _Peter Bala_, Oct 31 2021