|
%I #10 Jan 15 2018 15:33:12
%S 4,5,6,7,9,11,23,47,59,83,107,167,179,227,263,347,359,383,467,479,503,
%T 563,587,719,839,863,887,983,1019,1187,1283,1307,1319,1367,1439,1487,
%U 1523,1619,1823,1907,2027,2039,2063,2099,2207,2447,2459
%N Increasingly ordered list of those values m for which the degree of the minimal polynomial of 2*cos(Pi/m) (see A187360) is prime.
%C The degree delta(m) of the minimal polynomial of rho(m) := 2*cos(Pi/m), called C(m,x) with coefficient array A187360, is given by A055034(m).
%C If delta(m) = phi(2*m)/2, m>=2, delta(1) = 1, with phi = A000010, is prime then the (Abelian) Galois group G(Q(rho(m))/Q) is cyclic. Because this Galois group of C(m,x) has order delta(m) this follows from a corollary to Lagrange's theorem, or also from Cauchy's theorem on groups.
%C Because the mentioned Galois group is isomorphic to the multiplicative group Modd m of order delta(m) (see a comment on A203571) all m = a(n) values appear in A206551.
%C This sequence is also a subsequence of A210845 because p is squarefree (see A005117).
%F phi(2*m)/2 is prime iff m=a(n), n>=1, with phi = A000010 (Euler's totient).
%e a(4) = 7, because 7 satisfies phi(14)/2 = phi(2*7)/2 = 1*6/2 = 3, which is prime; and 7 is the fourth smallest number m satisfying: phi(2*m)/2 is prime.
%Y Cf. A055034.
%K nonn
%O 1,1
%A _Wolfdieter Lang_, Sep 03 2012
|
