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%I #36 May 14 2013 12:59:01
%S 3,11,41,2131,110771,15558008491
%N Primes p such that there is a prime q satisfying 3*p^2 - q^2 = 2.
%C Primes q: 5, 19, 71, 3691, 191861,...
%C (q - p)/2: 1, 4, 15, 780, 40545,...
%C a(7) > 2.8016852867294*10^4857. - _Zak Seidov_, May 09 2013
%C Probably finite.
%C This is a form of Pell's equation with the requirement that solutions be prime. - _T. D. Noe_, May 14 2013
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/PellEquation.html">MathWorld: Pell Equation</a>
%e 11 is prime and sqrt(3*11^2 - 2) = sqrt(361) = 19 is prime, so 11 is in the sequence.
%t nn = 1000; ta = LinearRecurrence[{4, -1}, {1, 3}, nn]; tb = LinearRecurrence[{4, -1}, {1, 5}, nn]; sol = Select[Range[nn], PrimeQ[ta[[#]]] && PrimeQ[tb[[#]]] &]; ta[[sol]] (* _T. D. Noe_, May 14 2013 *)
%o (PARI) v=[1,1]; for(i=1,1e4,v=[v[2],4*v[2]-v[1]]; if(ispseudoprime(v[2]) && ispseudoprime(sqrtint(3*v[2]^2-2)), print1(v[2]", "))) \\ _Charles R Greathouse IV_, May 13 2013
%o (PFGW)
%o ABC2 Linear(3,11,41,153,$a) & Linear(5,19,71,265,$a)
%o a: from 3 to 20000 // _Charles R Greathouse IV_, May 13 2013
%K nonn
%O 1,1
%A _Irina Gerasimova_, May 07 2013
%E a(4) from R. J. Mathar, May 07 2013
%E a(6) from Charles R Greathouse IV, May 07 2013
%E a(5) from Zak Seidov, May 09 2013