A348723 - OEIS

%I #13 Nov 08 2021 16:32:20

%S 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,

%T 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,

%U 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

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

%C 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 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.

%C 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.

%C 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).

%H 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 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 r_0 = 2*(cos(4*Pi/19) + cos(6*Pi/19) - cos(9*Pi/19)) + 1.

%F 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)).

%F r_0 = 1/(8*cos(4*Pi/19)*cos(6*Pi/19)*cos(9*Pi/19)).

%F 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) ).

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

%F r_0 = abs( (1 - z^4)*(1 - z^6)*(1 - z^9)/((1 - z)*(1 - z^7)*(1 - z^8)) ).

%F 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}.

%e 3.50701864409297629866079992371567802902597642013036...

%p 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);

%t 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 *)

%o (PARI) 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

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

%K nonn,cons,easy

%O 1,1

%A _Peter Bala_, Oct 31 2021