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Minimal polynomials of sin(2Pi/n) are mapped to those of cos(2Pi/a(n)).
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%I #21 Nov 03 2019 16:06:21

%S 4,4,12,1,20,12,28,8,36,20,44,6,52,28,60,16,68,36,76,5,84,44,92,24,

%T 100,52,108,14,116,60,124,32,132,68,140,9,148,76,156,40,164,84,172,22,

%U 180,92,188,48,196,100,204,13,212,108,220,56,228,116,236,30,244,124,252,64,260,132,268,17,276,140,284,72,292,148,300,38,308,156,316,80

%N Minimal polynomials of sin(2Pi/n) are mapped to those of cos(2Pi/a(n)).

%C The minimal polynomials of cos(2*Pi/n) are treated, e.g. in the Lehmer, Niven and Watkins-Zeitlin references. Lehmer and Niven call them psi_n(x) (eq. (1) and Lemma 3.8, p.37, respectively). In the latter reference they are called Psi_n(x), and we call them Psi(n,x). By definition (Niven, p. 28) these are monic, rational polynomials which have as a root cos(2*Pi/n) and are of minimal degree. They are irreducible (Niven p. 37, Lemma 3.8). See also A181875 for more details and a link with Psi(n,x), n=1..30.

%C The minimal polynomials of sin(2*Pi/n) are treated, e.g. in the Lehmer and Niven references. Lehmer's theorem 2 is, however, incorrect. See A181872 and the link there for a counterexample. In this link one can also find these polynomials, called Pi(n,x), for n=1..30.

%C The sequence a(n) translates these polynomials: Pi(n,x) = Psi(a(n),x), n >= 1. This translation is based on the trigonometric identity: sin(2*Pi/n) = cos(2*Pi*r(n)), with r(n):=|(4-n)/(4*n)|.

%C a(n):=denominator(r(n)) (in lowest terms). Note that the degrees agree with those given in the Niven reference, Theorem 3.9, p. 37.

%D I. Niven, Irrational Numbers, The Math. Assoc. of America, second printing, 1963, distributed by John Wiley and Sons.

%H Michael De Vlieger, <a href="/A178182/b178182.txt">Table of n, a(n) for n = 1..10000</a>

%H D. H. Lehmer, <a href="http://www.jstor.org/stable/2301023">A Note on Trigonometric Algebraic Numbers</a>, Am. Math. Monthly 40,3 (1933) 165-6.

%H Pinthira Tangsupphathawat, Takao Komatsu, Vichian Laohakosol, <a href="https://www.emis.de/journals/JIS/VOL21/Laohakosol/lao8.html">Minimal Polynomials of Algebraic Cosine Values, II</a>, J. Int. Seq., Vol. 21 (2018), Article 18.9.5.

%H W. Watkins and J. Zeitlin, <a href="http://www.jstor.org/stable/2324301">The Minimal Polynomial of cos(2Pi/n)</a>, Am. Math. Monthly 100,5 (1993) 471-4.

%F a(n) = denominator(|(n-4)/(4*n)|), n >= 1.

%F a(n) = 4*n/gcd(n-4,16). a(n) = 4*n if n is odd; if n is even then a(n) = 2*n if n/2 == 1, 3, 5, 7 (mod 8), a(n) = n if n/2 == 0, 4 (mod 8), a(n) = n/2 if n/2 == 6 (mod 8) and a(n) = n/4 if n/2 == 2 (mod 8). - _Wolfdieter Lang_, Dec 01 2013

%F a(2*n)/(2*n) = 1/4, 1/2, 1, and 2, for n == 2 (mod 8), 6 (mod 8), 0 (mod 4), and 1 (mod 2), for n >= 1. The reciprocal can be used in a formula for the zeros of the minimal polynomials of 2*sin(Pi/2) (A228786). See A327921. - _Wolfdieter Lang_, Nov 02 2019

%e Pi(5,x) = Psi(20,x) because sin(2*Pi/5) = cos(2*Pi/20).

%t Array[4 #/GCD[# - 4, 16] &, 80] (* _Michael De Vlieger_, Feb 07 2019 *)

%Y Cf. A181872, A228786, A327921.

%K nonn,easy

%O 1,1

%A _Wolfdieter Lang_, Jan 11 2011