A348728 -id:A348728 - cp's OEIS frontend

A348729 Decimal expansion of the positive root of Shanks's simplest cubic associated with the prime p = 163.

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

Offset: 2

Peter Bala, Nov 06 2021

Let a be a natural number 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 = 11, corresponding to the prime p = 163, the three real roots of Shanks' cubic x^3 - 11*x^2 - 14*x - 1 in descending order are r_0 = 12.1582466687..., r_1 = - -0.0759979672... and r_2 = -1.0822487014.... Here we consider the positive root r_1.
The linear fractional transformation z -> - 1/(1 + z) cyclically permutes the three roots r_0, r_1 and r_2: the quadratic mapping z -> z^2 - 12*z - 2 also cyclically permutes the roots.
The algebraic number field Q(r_0) is a totally real cubic field with class number 4 and discriminant equal to 163^2. 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.

			12.15824666871213538260037124700042984524658480470748 ...

D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152.
Index entries for algebraic numbers, degree 3.

Cf. A005471, A160389, A255240, A255241, A255249, A348720 - A348728.

R := convert([seq(mod(n^3, 163), n = 1..162)], set):
P := k -> sqrt( mul(sin((1/163)*k*r*Pi), r in R) ):
evalf(sqrt(P(3)/P(1)), 105);

rs = Union@Mod[Range[1, 162]^3, 163]; f[k_] := Sqrt[Product[Sin[k*r*Pi/163], {r, rs}]]; RealDigits[Sqrt[f[3]/f[1]], 10, 100][[1]] (* Amiram Eldar, Nov 08 2021 *)

polrootsreal(x^3 - 11*x^2 - 14*x - 1)[3] \\ Charles R Greathouse IV, Feb 04 2025

Let R = {1, 5, 6, 8, ..., 155, 157, 158, 162} denote the multiplicative subgroup of nonzero cubic residues in the finite field Z_163, with cosets 2*R = {2, 7, 9, 10, ..., 153, 154, 156, 161} and 3*R = {3, 4, 11, 14, ..., 149, 152, 159, 160}.
Define P(k) = Product_{r in R, r <= (163-1)/2} sin(k*r*Pi/163). The three roots of the cubic x^3 - 11*x^2 - 14*x - 1 are
r_0 =  sqrt(P(3)/P(1)) = 12.1582466687....
r_1 = -sqrt(P(1)/P(2)) = -0.0759979672....
r_2 = -sqrt(P(2)/P(3)) = -1.0822487014....

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

Original entry on oeis.org

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

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 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.

			5.34471236545183496316569141884698646995869587081422 ...

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

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(2)/R(3), 100);

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 *)

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).

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) ).

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A348729 Decimal expansion of the positive root of Shanks's simplest cubic associated with the prime p = 163.

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

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A348727 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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