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Odd primes satisfying a specific condition (see comments).
2

%I #13 Jun 05 2015 07:06:05

%S 19,23,163,487,1459,2663,39367,410759,715823,2450087,12872687,

%T 13935743,23394167,86093443,160125983,219804479,236741543,258280327,

%U 2116179719,3991233959,4715895383,6109873703,8487319319,9264815927,12601744847

%N Odd primes satisfying a specific condition (see comments).

%C Condition on odd prime p so that Q(Cp^2) is not rational over Q.

%C Let p>7 is an odd prime which does not satisfy any of the following conditions:

%C (i) p = 2*3^s + 1, s >=0 where s !== -1 modulo 4.

%C (ii) p = 2*11^(2s+1) + 1, s>=0.

%C (iii) p = 2*q^(2s+1) + 1, s>=1 where q is an odd prime such that q == -1 modulo mod 12, q >= 23.

%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 3.7 p.19).

%o (PARI) iscondi(p) = (r = (p-1)/2) && (k = ispower(r,,&n)) && (n == 3) && (k >= 2) && ((k % 4) != 3);

%o iscondii(p) = (r = (p-1)/2) && ((r == 11) || ((k = ispower(r,,&n)) && (n == 11) && (k % 2)));

%o iscondiii(p) = (r = (p-1)/2) && (k = ispower(r,,&n)) && isprime(n) && (n >= 23) && ((n % 12) == 11) && (k >= 3) && (k % 2);

%o isok(p) = isprime(p) && (iscondi(p) || iscondii(p) || iscondiii(p));

%Y Cf. A240583, A240584.

%K nonn,more

%O 1,1

%A _Michel Marcus_, Apr 08 2014