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Primes p such that there is a prime q satisfying 3*p^2 - q^2 = 2.
1

%I #37 Feb 16 2025 08:33:19

%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="https://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,changed

%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