A178182 Minimal polynomials of sin(2Pi/n) are mapped to those of cos(2Pi/a(n)).

4, 4, 12, 1, 20, 12, 28, 8, 36, 20, 44, 6, 52, 28, 60, 16, 68, 36, 76, 5, 84, 44, 92, 24, 100, 52, 108, 14, 116, 60, 124, 32, 132, 68, 140, 9, 148, 76, 156, 40, 164, 84, 172, 22, 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

Offset: 1

Author

Wolfdieter Lang, Jan 11 2011

Keywords

• nonn
• easy

Comments

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.

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.

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)|.

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.

Examples

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

References

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

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• D. H. Lehmer, A Note on Trigonometric Algebraic Numbers, Am. Math. Monthly 40,3 (1933) 165-6.
• Pinthira Tangsupphathawat, Takao Komatsu, Vichian Laohakosol, Minimal Polynomials of Algebraic Cosine Values, II, J. Int. Seq., Vol. 21 (2018), Article 18.9.5.
• W. Watkins and J. Zeitlin, The Minimal Polynomial of cos(2Pi/n), Am. Math. Monthly 100,5 (1993) 471-4.

Crossrefs

Cf. A181872, A228786, A327921.

Programs

Mathematica

Array[4 #/GCD[# - 4, 16] &, 80] (* Michael De Vlieger, Feb 07 2019 *)

Formula

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

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

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