A180634 - OEIS

%I #36 Oct 29 2023 17:28:08

%S 1,2,8,12,15,16,20,21,24,28,30,32,33,35,36,39,40,42,44,45,48,51,52,55,

%T 56,57,60,63,64,65,66,68,69,70,72,75,76,77,78,80,84,85,87,88,90,91,92,

%U 93,95,96,99,100

%N Numbers n such that the discriminant of the n-th cyclotomic polynomial is a square.

%C A number n is in this sequence if the Galois group of the n-th cyclotomic polynomial over the rationals contains only even permutations.

%C Essentially the same as A033949. - _R. J. Mathar_, Oct 15 2011

%C Also, numbers n such that the product of the elements in the group Z_n of invertible elements mod n (i.e., the product mod n of x such that 1 <= x < n and x is coprime to n) is 1. An equivalent characterization of the latter (apart from n=2): n such that the number of square roots of 1 mod n is divisible by 4. (See comments at A033949). - _Robert Israel_, Dec 08 2014

%C To see this, use Gauss's generalization of Wilson's theorem namely, the product of the units of Z_n is -1 if n is 4 or p^i or 2p^i for odd primes p, i >0,  and is equal to 1 otherwise. - _W. Edwin Clark_, Dec 09 2014

%H Robert G. Wilson v, <a href="/A180634/b180634.txt">Table of n, a(n) for n = 1..1000</a>

%H Mohammad K. Azarian, <a href="http://www.ijpam.eu/contents/2007-36-2/9/9.pdf">On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials</a>, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257.  Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.

%e n=5: The 5th cyclotomic polynomial is x^4+x^3+x^2+x+1 with discriminant 125, which is not a square. The Galois group is generated by (1243), that is an odd permutation. Hence 5 is not in the sequence. n=8: The 8th cyclotomic polynomial is x^4+1 with discriminant 256, which is a square. The Galois group is {id,(13)(57),(15)(37),(17)(35)}, that are all even permutations. Hence 8 is in the sequence.

%p m := proc(n) local k, r; r := 1;

%p for k from 1 to n do if igcd(n,k) = 1 then r := modp(r*k,n) fi od; r end:

%p [1, op(select(n -> m(n) = 1, [$1..100]))]; # _Peter Luschny_, May 25 2017

%t fQ[n_] := IntegerQ@ Sqrt@ Discriminant[ Cyclotomic[ n, x], x]; Select[ Range@ 100, fQ] (* _Robert G. Wilson v_, Dec 10 2014 *)

%o (PARI) for(n=1,100,if(issquare(poldisc(polcyclo(n))),print(n)))

%Y Cf. A004124, A033949.

%K easy,nonn

%O 1,2

%A _Jan Fricke_, Sep 13 2010