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

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, 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, 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

Offset: 1

Author

Peter Bala, Oct 31 2021

Keywords

• nonn
• cons
• easy

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 (r_0) and A348724 (|r_1|).

Here we consider the absolute value of the root r_2.

Examples

			1.28514248182978536439411987353062741342678092572261 ...
		

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 (see case 9 in the table)
• D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152

Crossrefs

Cf. A005471, A160389, A255240, A255241, A255249, A348720 - A348729.

Programs

Maple

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);

Mathematica

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 *)

Formula

|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)).

|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) ).

|r_2| = 2*(cos(Pi/19) + cos(7*Pi/19) - cos(8*Pi/19)) - 1.

Let z = exp(2*Pi*i/19). Then

|r_2| = abs( (1 - z)*(1 - z^7)*(1 - z^8)/((1 - z^2)*(1 - z^3)*(1 - z^5)) ).

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)^(1/19) + (-1)^(7/19) - (-1)^(8/19) + (-1)^(11/19) - (-1)^(12/19) - (-1)^(18/19). - Peter Luschny, Nov 08 2021