A348728 - OEIS

%I #15 Nov 08 2021 16:32:58

%S 1,1,8,7,1,0,0,8,0,7,6,0,6,4,0,9,2,0,1,6,8,3,3,7,0,0,9,8,7,2,2,7,6,1,

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

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

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

%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. The polynomial has three real roots, one positive and two negative.

%C 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....

%C Here we consider the absolute value of the root r_2. See A348726 (r_0) and A348727 (|r_1|) for the other two roots.

%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 (see case 37 in the table)

%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_2| = 2*(-cos(Pi/37) + cos(6*Pi/37) + cos(8*Pi/37) + cos(10*Pi/37) - cos(11*Pi/37) + cos(14*Pi/37)) - 1.

%F |r_2| = |R(3)/R(1)|, 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 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 the constant equals Product_{n >= 0} ( Product_{k in the coset 3*R} (37*n+k) )/( Product_{k in the group R} (37*n + k) ).

%e 1.18710080760640920168337009872276109935284715168366...

%p 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(3)/R(1), 100);

%t f[ks_,m_] := Product[Sin[k*Pi/m], {k,ks}]; ks = {1, 6, 8, 10, 11, 14}; RealDigits[f[3*ks,37]/f[ks,37], 10, 100][[1]] (* _Amiram Eldar_, Nov 08 2021 *)

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

%K nonn,cons,easy

%O 1,3

%A _Peter Bala_, Oct 31 2021