A348725 -id:A348725 - cp's OEIS frontend

A348723 Decimal expansion of the positive root of Shanks' simplest cubic associated with the prime p = 19.

3, 5, 0, 7, 0, 1, 8, 6, 4, 4, 0, 9, 2, 9, 7, 6, 2, 9, 8, 6, 6, 0, 7, 9, 9, 9, 2, 3, 7, 1, 5, 6, 7, 8, 0, 2, 9, 0, 2, 5, 9, 7, 6, 4, 2, 0, 1, 3, 0, 3, 6, 9, 6, 7, 5, 1, 2, 6, 5, 8, 2, 1, 7, 8, 3, 5, 2, 9, 7, 6, 9, 6, 4, 8, 2, 1, 0, 1, 9, 9, 7, 1, 5, 7, 6, 0, 0, 3, 4, 0, 8, 6, 1, 9, 4, 0, 9, 0, 7, 1, 5

Offset: 1

Peter Bala, Oct 31 2021

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 A348724 and A348725.
The quadratic mapping z -> z^2 - 3*z - 2 cyclically permutes the roots of the cubic: the mapping z -> - z^2 + 2*z + 4 gives the inverse cyclic permutation of the three roots.
The algebraic number field Q(r_0) is a totally real cubic field of class number 1 and discriminant equal to 19^2. The Galois group of Q(r_0) over Q is a cyclic group of order 3. The real numbers r_0 and 1 + r_0 are two independent fundamental units of the field Q(r_0). See Shanks. In Cusick and Schoenfeld, r_0 and r_1 (denoted there by E_1 and E_2) are taken as a fundamental pair of units (see case 9 in the table).

			3.50701864409297629866079992371567802902597642013036...

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

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

evalf(sin(4*Pi/19)*sin(6*Pi/19)*sin(9*Pi/19)/(sin(Pi/19)*sin(7*Pi/19)*sin(8*Pi/19)), 100);

RealDigits[Sin[4*Pi/19]*Sin[6*Pi/19]*Sin[9*Pi/19]/(Sin[Pi/19]*Sin[7*Pi/19]*Sin[8*Pi/19]), 10, 100][[1]] (* Amiram Eldar, Nov 08 2021 *)

sin(4*Pi/19)*sin(6*Pi/19)*sin(9*Pi/19)/(sin(Pi/19)*sin(7*Pi/19)*sin(8*Pi/19)) \\ Michel Marcus, Nov 08 2021

r_0 = 2*(cos(4*Pi/19) + cos(6*Pi/19) - cos(9*Pi/19)) + 1.
r_0 = sin(4*Pi/19)*sin(6*Pi/19)*sin(9*Pi/19)/(sin(Pi/19)*sin(7*Pi/19)*sin(8*Pi/19)).
r_0 = 1/(8*cos(4*Pi/19)*cos(6*Pi/19)*cos(9*Pi/19)).
r_0 = Product_{n >= 0} (19*n+4)*(19*n+6)*(19*n+9)*(19*n+10)*(19*n+13)*(19*n+15)/( (19*n+1)*(19*n+7)*(19*n+8)*(19*n+11)*(19*n+12)*(19*n+18) ).
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) ).
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) ).
Let z = exp(2*Pi*i/19). Then
r_0 = abs( (1 - z^4)*(1 - z^6)*(1 - z^9)/((1 - z)*(1 - z^7)*(1 - z^8)) ).
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}.

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

Original entry on oeis.org

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, 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, 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

Offset: 0

Peter Bala, Oct 31 2021

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.

			0.22187616226319093426668005018505061559919549440775...

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

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

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);

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 *)

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

|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

cp's OEIS Frontend

A348723 Decimal expansion of the positive root of Shanks' simplest cubic associated with the prime p = 19.

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