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%I #14 Sep 23 2018 08:56:47
%S 47,79,167,191,223,239,263,359,367,383,431,439,463,479,503,599,607,
%T 719,823,839,863,887,911,983,1031,1039,1087,1103,1223,1231,1303,1319,
%U 1327,1367,1399,1439,1447,1487,1511,1543,1559,1583,1663,1759,1823,1831,1847
%N Odd primes satisfying a specific condition (see comments).
%C Condition on odd prime p so that Q(Cp) is not rational over Q: p = 2q+1 where q = -1 (mod 4), q is squarefree, and any of 4p-q and q+1 is not square.
%H Shizuo Endo and Takehiko Miyata, <a href="http://dx.doi.org/10.2969/jmsj/02510007">Invariants of finite abelian groups</a>, J. Math. Soc. Japan, Volume 25, Number 1 (1973), 1-167 (see Proposition 3.6 (i) p.18).
%H Akinari Hoshi, <a href="http://arxiv.org/abs/1402.3678">On Noether's problem for cyclic groups of prime order</a>, arXiv:1402.3678 [math.NT], 2014 (see Proposition 3.1 (i) p.4 and Table 1 p.19).
%t Select[Prime[Range[2, 300]], (q = (#-1)/2; SquareFreeQ[q] && Mod[q, 4] == 3 && !IntegerQ[Sqrt[4#-q]] && !IntegerQ[Sqrt[q+1]])&] (* _Jean-François Alcover_, Sep 22 2018 *)
%o (PARI) isok(p) = isprime(p) && ((q = (p-1)/2) && issquarefree(q) && ((q % 4) == 3) && (!issquare(4*p-q)));
%Y Cf. A240584, A240585.
%K nonn
%O 1,1
%A _Michel Marcus_, Apr 08 2014
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