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

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, 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, 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

Offset: 1

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 = 5.3447123654..., 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.

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).

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}.

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

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....

The linear fractional transformation z -> - 1/(1 + z) cyclically permutes the three roots of the cubic polynomial.

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.

Examples

			5.34471236545183496316569141884698646995869587081422 ...
		

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
• 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(2)/R(3), 100);

Mathematica

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 *)

Formula

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)).

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) );

|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) );

|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) ).

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).