A240584 - OEIS

%I #8 Dec 15 2018 08:34:43

%S 113,137,233,521,593,617,809,977,1033,1097,1129,1193,1289,1361,1489,

%T 1553,1609,1777,1993,2129,2153,2281,2417,2441,2473,2609,2729,2833,

%U 2897,3049,3089,3121,3209,3217,3433,3593,3761,3793,3881,4073,4241,4273,4297,4337

%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 = 8q+1 where q != -1 (mod 4), q is squarefree, and any of p-q and p-4q 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 (ii) 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 (ii) p.4 and Table 2 p.19).

%t Reap[For[p = 3, p < 5000, p = NextPrime[p], If[Mod[p, 8] == 1 && Mod[q = Quotient[p, 8], 4] != 3 && SquareFreeQ[q] && AllTrue[{p-q, p-4q}, !IntegerQ[Sqrt[#]]&], Sow[p]]]][[2, 1]] (* _Jean-François Alcover_, Dec 15 2018 *)

%o (PARI) isok(p) = isprime(p) &&  ((type(q = (p-1)/8)== "t_INT") && ((q % 4) != 3) && (!issquare(p-q)) && (!issquare(p-4*q)));

%Y Cf. A240583, A240585.

%K nonn

%O 1,1

%A _Michel Marcus_, Apr 08 2014