OFFSET
0,1
COMMENTS
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.
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.
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.
LINKS
T. W. Cusick and Lowell Schoenfeld, A table of fundamental pairs of units in totally real cubic fields, Math. Comp. 48 (1987), 147-158
D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152
FORMULA
|r_1| = 2*(cos(3*Pi/19) + cos(5*Pi/19) - cos(2*Pi/19)) - 1.
|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)).
|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) ).
Let z = exp(2*Pi*i/19). Then
|r_1| = abs( (1 - z^2)*(1 - z^3)*(1 - z^5)/((1 - z^4)*(1 - z^6)*(1 - z^9)) ).
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}.
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
EXAMPLE
0.22187616226319093426668005018505061559919549440775...
MAPLE
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);
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
AUTHOR
Peter Bala, Oct 31 2021
STATUS
approved