The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A055035 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing entries 1-10 | older changes

A055035

Degree of minimal polynomial of sin(Pi/n) over the rationals.
(history; published version)

#43 by Joerg Arndt at Mon Jan 16 02:39:39 EST 2023

STATUS

editing

approved

#42 by Joerg Arndt at Mon Jan 16 02:39:37 EST 2023

MATHEMATICA

fa[n_] := Exponent[ MinimalPolynomial[ Sin[ Pi/n]][x], x]; Array[f, a, 75] (* Robert G. Wilson v, Jul 28 2014 *)

STATUS

approved

editing

#41 by N. J. A. Sloane at Wed Feb 02 23:33:24 EST 2022

STATUS

proposed

approved

#40 by Michael De Vlieger at Wed Feb 02 20:35:46 EST 2022

STATUS

editing

proposed

#39 by Michael De Vlieger at Wed Feb 02 20:35:44 EST 2022

LINKS

Sameen Ahmed Khan, <a href="https://doi.org/10.13189/ms.2021.090605">Trigonometric Ratios Using Algebraic Methods</a>, Mathematics and Statistics (2021) Vol. 9, No. 6, 899-907.

STATUS

approved

editing

#38 by Bruno Berselli at Thu Oct 31 05:29:49 EDT 2019

STATUS

proposed

approved

#37 by Michel Marcus at Thu Oct 31 04:36:32 EDT 2019

STATUS

editing

proposed

#36 by Michel Marcus at Thu Oct 31 04:36:21 EDT 2019

NAME

Degree of minimal polynomial of sin(piPi/n) over the rationals.

COMMENTS

Also degree of minimal polynomial of function F(n)=(gamma(1/n)*gamma((n-1)/n))/pi Pi over the rationals. Roots of minimal polynomials of F(n) belonging to algebraic extension of sin(n/Pi) and vice versa (e.g. gamma(1/11)*gamma(10/11)/pi Pi = 20*sin(piPi/11) - 112*sin(piPi/11)^3 + 256*sin(piPi/11)^5 - 256*sin(piPi/11)^7 + (1024*sin(piPi/11)^9)/11). - Artur Jasinski, Oct 17 2011

The algebraic numbers sin(piPi/(2*l)) are given in A228783 in the power basis of the number field Q(2*cos(piPi/(2*l))) if n is even and of Q(2*cos(piPi/l)) if l is odd. In A228785 , sin(piPi/(2*l+1)) is given in the power basis of Q(2*cos(piPi/(2*(2*l+1)))) (only odd powers appear). The minimal polynomials for 2*sin(piPi/n), n>=1, are given in A228786. - Wolfdieter Lang, Oct 10 2013

STATUS

approved

editing

Discussion

Thu Oct 31

04:36

Michel Marcus: Pi rather than pi

#35 by Wolfdieter Lang at Tue Oct 29 17:57:06 EDT 2019

STATUS

editing

approved

#34 by Wolfdieter Lang at Tue Oct 29 17:56:54 EDT 2019

FORMULA

a(1)=1, a(2)=1, a(n)=phi(n)/(1, 1, 2, 1 for n=0, 1, 2, 3 mod 4) for n>2, where phi is Euler's totient, A000010.

a(n) = A093819(2*n), n >= 1.- Wolfdieter Lang, Oct 29 2019

CROSSREFS

Cf. A000010, A228786 (row length), A093819.

STATUS

approved

editing