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

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, 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, 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

Offset: 0

Author

Peter Bala, Oct 31 2021

Keywords

• nonn
• cons
• easy

Comments

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.

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.

Examples

			0.15761155784542576148232132012422537060584871913055 ...
		

Links

• T. W. Cusick and Lowell Schoenfeld, A table of fundamental pairs of units in totally real cubic fields, Math. Comp. 48 (1987), 147-158 (see case 37 in the table)
• D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152

Crossrefs

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

Programs

Maple

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);

Mathematica

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 *)

Formula

|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)).

|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.

|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).

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

Product_{n >= 0} ( Product_{k in the coset 2*R} (37*n+k) )/( Product_{k in the group R} (37*n + k) ).