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%I #12 May 23 2022 22:33:19
%S 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,
%T 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,
%U 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
%N Decimal expansion of the positive root of Shanks's simplest cubic associated with the prime p = 163.
%C 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.
%C 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.
%C 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.
%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 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}.
%F 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
%F r_0 = sqrt(P(3)/P(1)) = 12.1582466687....
%F r_1 = -sqrt(P(1)/P(2)) = -0.0759979672....
%F r_2 = -sqrt(P(2)/P(3)) = -1.0822487014....
%e 12.15824666871213538260037124700042984524658480470748 ...
%p R := convert([seq(mod(n^3, 163), n = 1..162)], set):
%p P := k -> sqrt( mul(sin((1/163)*k*r*Pi), r in R) ):
%p evalf(sqrt(P(3)/P(1)), 105);
%t 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 *)
%Y Cf. A005471, A160389, A255240, A255241, A255249, A348720 - A348728.
%K nonn,cons,easy
%O 2,2
%A _Peter Bala_, Nov 06 2021
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