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A348726 Decimal expansion of the positive root of Shanks' simplest cubic associated with the prime p = 37.

%I A348726 #17 Nov 08 2021 16:32:44
%S A348726 5,3,4,4,7,1,2,3,6,5,4,5,1,8,3,4,9,6,3,1,6,5,6,9,1,4,1,8,8,4,6,9,8,6,
%T A348726 4,6,9,9,5,8,6,9,5,8,7,0,8,1,4,2,2,4,9,4,6,3,9,6,3,6,1,7,5,6,0,1,5,4,
%U A348726 5,3,8,5,7,2,1,1,5,7,7,0,1,2,1,6,8,7,6,6,8,2,1,9,1,4,2,4,3,4,1,6,9
%N A348726 Decimal expansion of the positive root of Shanks' simplest cubic associated with the prime p = 37.
%C A348726 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 A348726 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 = 5.3447123654..., r_1 = - 0.1576115578... and r_2 = - 1.1871008076.... Here we consider the positive root r_0. See A348727 (|r_1|) and A348728 (|r_2|) for the other two roots.
%C A348726 The algebraic number field Q(r_0) is a totally real cubic field with class number 1 and discriminant equal to 37^2. 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 37 in the table).
%C A348726 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}.
%C A348726 Define 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). Then the three roots of the cubic x^3 - 4*x^2 - 7*x - 1 are
%C A348726 r_0 = - R(2)/R(3) = 5.3447123654..., r_1 = - R(1)/R(2) = - 0.1576115578... and r_2 = R(3)/R(1) = - 1.1871008076....
%C A348726 The linear fractional transformation z -> - 1/(1 + z) cyclically permutes the three roots of the cubic polynomial.
%C A348726 The quadratic mapping z -> z^2 - 5*z - 2 also cyclically permutes the roots of the cubic: the inverse cyclic permutation of the roots is given by z -> - z^2 + 4*z + 6.
%H A348726 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 A348726 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 A348726 r_0 = 1 + 2*(cos(3*Pi/37) - cos(4*Pi/37) + cos(5*Pi/37) + cos(7*Pi/37) + cos(13*Pi/37) - cos(18*Pi/37)).
%F A348726 r_0 = |R(2)/R(3)| = Product_{n >= 0} ( Product_{k in the coset 2*R} (37*n+k) )/( Product_{k in the coset 3*R} (37*n + k) );
%F A348726 |r_1| = |R(1)/R(2)| = Product_{n >= 0} ( Product_{k in the group R} (37*n+k) )/( Product_{k in the coset 2*R} (37*n + k) );
%F A348726 |r_2| = |R(3)/R(1)| = Product_{n >= 0} ( Product_{k in the coset 3*R} (37*n+k) )/( Product_{k in the group R} (37*n + k) ).
%F A348726 R(2)/R(1) + R(2)/R(3) = 1 = R(3)/R(2) - R(3)/R(1) = R(1)/R(2) - R(1)/R(3).
%e A348726 5.34471236545183496316569141884698646995869587081422 ...
%p A348726 R := k -> sin(k*Pi/37)*sin(6*k*Pi/37)*sin(8*k*Pi/37)*
%p A348726 sin(10*k*Pi/37)*sin(11*k*Pi/37)*sin(14*k*Pi/37): evalf(-R(2)/R(3), 100);
%t A348726 f[ks_,m_] := Product[Sin[k*Pi/m], {k,ks}]; ks = {1, 6, 8, 10, 11, 14}; RealDigits[f[2*ks,37]/f[3*ks,37], 10, 100][[1]] (* _Amiram Eldar_, Nov 08 2021 *)
%Y A348726 Cf. A005471, A160389, A255240, A255241, A255249, A348720 - A348729.
%K A348726 nonn,cons,easy
%O A348726 1,1
%A A348726 _Peter Bala_, Oct 31 2021