A348726 -id:A348726 - cp's OEIS frontend

A348728 Decimal expansion of the absolute value of one of the negative roots of Shanks' simplest cubic associated with the prime p = 37.

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, 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, 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

Offset: 1

Peter Bala, Oct 31 2021

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

			1.18710080760640920168337009872276109935284715168366...

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

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

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);

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 *)

|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.
|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).
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) ).

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

Original entry on oeis.org

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

Peter Bala, Oct 31 2021

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.

			0.15761155784542576148232132012422537060584871913055 ...

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

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

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);

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 *)

|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) ).

cp's OEIS Frontend

A348728 Decimal expansion of the absolute value of one of the negative roots of Shanks' simplest cubic associated with the prime p = 37.

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