cp's OEIS Frontend

A348727 Decimal expansion of the absolute value of one of the negative roots of Shanks' simplest cubic associated with the prime p = 37.

%I A348727 #14 Nov 08 2021 16:32:51
%S A348727 1,5,7,6,1,1,5,5,7,8,4,5,4,2,5,7,6,1,4,8,2,3,2,1,3,2,0,1,2,4,2,2,5,3,
%T A348727 7,0,6,0,5,8,4,8,7,1,9,1,3,0,5,5,9,9,3,0,3,6,8,4,9,1,3,0,5,4,1,7,0,9,
%U A348727 6,0,5,3,1,4,9,3,3,6,4,6,6,5,1,8,1,8,3,0,6,2,1,0,4,2
%N A348727 Decimal expansion of the absolute value of one of the negative roots of Shanks' simplest cubic associated with the prime p = 37.
%C A348727 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. The polynomial has three real roots, one positive and two negative.
%C A348727 In the case a = 4, corresponding to the prime p = 37, the three real roots of the cubic x^3 - 4*x^2 - 7*x - 1 in descending order are r_0 = 5.344712365..., r_1 = - 0.1576115578... and r_2 = - 1.187100807.... Here we consider the absolute value of the root r_1 (|E_2| in the notation of Cusick and Schoenfeld). See A348726 (r_0) and A348728 (|r_2|) for the other two roots.
%H A348727 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 (see case 37 in the table)
%H A348727 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 A348727 |r_1| = 1/((2^6)*cos(Pi/37)*cos(6*Pi/37)*cos(8*Pi/37)*cos(10*Pi/37)*cos(11*Pi/37)* cos(14*Pi/37)).
%F A348727 |r_1| = 2*(cos(2*Pi/37) - cos(9*Pi/37) + cos(12*Pi/37) - cos(15*Pi/37) + cos(16*Pi/37) - cos(17*Pi/37)) - 1.
%F A348727 |r_1| = R(1)/R(2), where R(k) = sin(k*Pi/37)*sin(6*k*Pi/37)* sin(8*k*Pi/37)*sin(10*k*Pi/37)*sin(11*k*Pi/37)*sin(14*k*Pi/37).
%F A348727 Let R = <1, 6, 8, 10, 11, 14, 23, 26, 27, 29, 31, 36> denote the multiplicative subgroup of nonzero cubic residues in the finite field Z_37, with cosets 2*R = {2, 9, 12, 15, 16, 17, 20, 21, 22, 25, 28, 35} and 3*R = {3, 4, 5, 7, 13, 18, 19, 24, 30, 32, 33, 34}. Then constant equals
%F A348727 Product_{n >= 0} ( Product_{k in the coset 2*R} (37*n+k) )/( Product_{k in the group R} (37*n + k) ).
%e A348727 0.15761155784542576148232132012422537060584871913055 ...
%p A348727 R := k -> sin(k*Pi/37)*sin(6*k*Pi/37)*sin(8*k*Pi/37)*sin(10*k*Pi/37)* sin(11*k*Pi/37)*sin(14*k*Pi/37): evalf(R(1)/R(2), 100);
%t A348727 f[ks_,m_] := Product[Sin[k*Pi/m], {k,ks}]; ks = {1, 6, 8, 10, 11, 14}; RealDigits[f[ks,37]/f[2*ks,37], 10, 100][[1]] (* _Amiram Eldar_, Nov 08 2021 *)
%Y A348727 Cf. A005471, A160389, A255240, A255241, A255249, A348720 - A348729.
%K A348727 nonn,cons,easy
%O A348727 0,2
%A A348727 _Peter Bala_, Oct 31 2021